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the area of a square is equal to s*s = s^2
s = 1 side of the square.
the area of the square is equal to x^2 + 6x + 9
this means that s^2 = x^2 + 6x + 9
this means that s = sqrt(x^2 + 6x + 9)
it just so happens that x^2 + 6x + 9 is a perfect square.
it is equal to (x+3)^2
this means that s^2 = (x+3)^2 which means that:
sqrt(s^2) = +/- sqrt(x+3)^2 which means that:
s = +/- (x+3)
if s = + (x+3) then:
s = x + 3
if s = -(x+3) then:
s = -x - 3
now s has to be > 0.
this means that:
x+3 > 0 which means that x > -3
this also means that:
-x-3 > 0 which means that x < -2
this means that x can be any real number except -3.
when x is equal to -3, s is equal to 0 and the area of the square is equal to 0 which means that you don't have a square of any magnitude.
the length of each side of your square is dependent on the value of x.
a graph of the equation of x^2 + 6x + 9 is shown below:
this graph shows the area of the square based on the value of x.

regardless of the value of x, this graph will be positive which means the area of the square will be positive.
this is everywhere except when x = -3.
when x = -3, the area is zero which we already indicated is not valid.
each side of your square, however, is not fixed to any one length.
the length of each side is dependent on the value of x.
a graph of the equation of y = +/- sqrt(x^2 + 6x + 9)}}} is shown below:
this graph is showing the value of the side of the square based on the value of x.

you can see that the length of each side will always be positive.
it's low point is 0 when x = -3, which is disallowed.
you asked me to draw your square.
any square will do.
just draw a rectangle which has all sides equal to each other.
each side of this square will be equal to sqrt(x^2 + 6x + 9).
x can be any value except -3.
you can also show the value of each side of the square as follows:
s = absolute value of (x+3).
in formula terms, this looks like:
s = |x+3|
when (x+3) is positive, you get s = x + 3
when (x+3) is negative, you get s = -(x + 3) which becomes s = -x - 3.
these are the same values we calculated above.
as ugly as it might look, that's your answer.
here's a picture of your square:
the length of each side in the square is equal to |x+3|.
this means each side in the square is equal to the absolute value of (x+3).

an example:
let x = -5.
each side of your square is equal to |-5+3| which is equal to |-2| which is equal to 2.
the area of your square is equal to x^2 + 6x + 9 after you replace x with (-5) to get (-5)^2 + 6*(-5) + 9 which is equal to 25 - 30 + 9 which is equal to 4.
since 2^4 = 4, the dimensions of each side and the area of the square confirm that the value of x = |-2| is good.
s is equal to 2
s^2 is equal to 4

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here's a reference that describes how it's done.
http://www.investorwords.com/tips/271/calculating-your-portfolios-beta.html
if they're correct, then all you are doing is calculating the average of all your betas based on the relative value of each stock in that portfolio.
you have 20 stocks in your portfolio.
each stock in the portfolio is worth 100,000 dollars.
the portfolio has a beta of 1.1
that would be the average of the beta of all the stocks in your portfolio.
you will be selling one stock that has a beta of .9 and using the proceeds to purchase another stock with a beta of 1.4.
the impact on your portfolios beta should be as follows:
the $100,000 stock is worth 100,000 / 2,000,000 = .05 of your portfolio
your portfolio's beta is 1.1
you subtract .05 * .9 from it and you add .05 * 1.4 to it.
the resulting impact on the portfolio beta should be:
1.1 - .05*.9 + .05*1.4 which becomes:
1.1 - .045 + .07 which becomes:
1.125
your portfolios beta after the transaction is equal to 1.125

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you are given the length of the base of the isosceles triangle.
with only this information, you can construct a limitless number of isosceles triangles.
check out the attached picture to see what I mean.
BC is the base.
AB and AC are the congruent sides.
AD is a perpendicular dropped from angle A intersecting with the base at D.
since this is an isosceles triangle, AD splits the base into 2 equal parts.
Those are BD and CD which are each equal to 7.
let y equal the length of AB and the length of AC
leg x equal the length of AD.
Since AD is perpendicular to the base, it forms 90 degree angles with the base.
Angles ADB and ADC are each 90 degrees.
This forms 2 right triangles.
They are triangle ADB and triangle ADC.
Since this is an isosceles triangle, angles B and C are equal to each other (they are congruent).
we can use the pythagorean formula to find the length of AD and the length of AC, but only in terms of each other since we don't have enough information to narrow the problem down to a specific result.
the pythagorean formula is:
hypotenuse squared = first leg squared plus second leg squared.
we'll call AD the first leg of right triangle ADC
we'll call DC the second leg of right triangle ADC
we'll call AC the hypotenuse of right triangle ADC
the length of AD is equal to x
the length of DC is equal to 7
the length of AC is equal to y
our formula becomes:
x^2 + 7^2 = y^2
this becomes:
x^2 + 49 = y^2
if we want to solve for the height of the triangle, this formula becomes:
x^2 = y^2 - 49
if we want to solve for the side of the triangle, this formula becomes:
y^2 = x^2 + 49 (it is left as is).
in order to find y, we need to know x.
in order to find x, we need to know y.
we'll assume we want to find y which is the congruent leg of the isosceles triangle.
the formula is:
y^2 = x^2 + 49 which becomes:
y = sqrt(x^2 + 49)
the perimeter of our isosceles triangle will be equal to:
p = 2y + 14 (p stands for perimeter.)
but...
we need to know x in order to find y.
since we can find any number of values for x, this means that our isosceles triangle will have any number of values for y which means that our isosceles triangle will have any number of values for p.
here's a table that lists a few values of y and p based on specified values of x.

DC	AD	AC	        perimeter
7	1	7.071067812	21.14213562
7	2	7.280109889	21.56021978
7	3	7.615773106	22.23154621
7	4	8.062257748	23.1245155
7	5	8.602325267	24.20465053
7	6	9.219544457	25.43908891
7	7	9.899494937	26.79898987
7	8	10.63014581	28.26029163
7	9	11.40175425	29.8035085
7	10	12.20655562	31.41311123
7	11	13.03840481	33.07680962
7	12	13.89244399	34.78488798
7	13	14.76482306	36.52964612
7	14	15.65247584	38.30495168

the length of DC is always 7.
the length of AD is equal to x in the diagram, but is equal to the specified values under the heading of AD in this table.
the length of AC is equal to y in the diagram, but is equal to the result of the equation y = sqrt(49+x^2) in this table.
the perimeter is equal to 2 times the length of AC plus the length of BC which is always equal to 14 because that's what was given.
you can see that you can have an infinite number of isosceles triangle which will create an infinite number of perimeters if the only information you have is that the base of the isosceles triangle is equal to 14.
your answer is:
without any other information other than the length of the base of the isosceles triangle, an infinite number of isosceles triangles can be drawn with a different altitude and a different length for the other 2 sides of the triangle.
in order to find the perimeter of one of the triangle, you have to be provided with either:
the height of the triangle.
the length of the congruent legs of the triangle.
the measure of the angle that is opposite the base of the triangle.
the measure of the congruent angles of the triangle.
-----
each of these would be in addition to the length of the base.
-----
note:
in an isosceles triangle:
the angles formed by the intersection of the legs of the triangle and the baswe of the triangle are congruent.
that would be angles B and C in the diagram.
the sides opposite these angles are congruent.
that would be sides AB and AC in the diagram.
here's your diagram

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L = length
W = width
L = W + 3
A = area
P = perimeter
P = 2L + 2W
A = L*W
The area has to be greater than 180 square meters.
This means that:
L*W >= 180
The perimeter has to be smaller than or equal to 70 meters.
This means that:
2L + 2W <= 70
the values for L and W that you are looking for will have to be able to satisfy both equations simultaneously.
Since L = W+3, we can substitute for L in these equation to get:
W^2 + 3W >= 180
4W + 6 <= 70
From the second equation, we get:
W <= 16
From the first equation, we get:
(W+15)*(W-12) >= 0
This means that (W+15) and (W-12) either both have to be positive or both have to be negative.
If both are positive, then W would have to be >= 12 because W-12 would be equal to 0 or less than 0 otherwise.
If both are negative, then W would have to be <= -15 because W + 15 would be positive otherwise.
So we have a situation where:
W >= 12 or W <= -15.
Since W could never be negative, then we have a situation where:
W would have to be greater than or equal to 12.
From the perimeter equation W needs to be less than or equal to 16.
From the area equation W needs to be greater than or equal to 12.
That means:
12 <= W <= 16
There are 6 possible values for W.
They are:
12
13
14
15
16
Since L is equal to W + 3, then there are 4 possible values of L corresponding to the 4 possible values of W.
The value pairs are:

     W        L
     12       15
     13       16
     14       17
     15       18
     16       19

These value pairs would result in a Perimeter and an Area of:

     W        L        P        A
     12       15       54       180
     13       16       58       208
     14       17       62       238
     15       18       66       270
     16       19       70       304

Assuming W = 11, then L = 14 and P = 50 and A = 154 which is less than 180 which means that W can't be less than 12 because the area requirements would be violated.
Assuming W = 17, then L = 20 and P = 74 which is greater than 70 which means that W can't be greater than 16 because the perimeter requirements would be violated.
the answers to your questions are:
what are the smallest dimensions that the garden could have?
12 by 15
What would the largest that it could have?
16 by 19
you can actually graph these equations and visually see where the dimensions have to be.
we'll do it for your own information because some problems in the future will probably require graphing to solve them.
the perimeter equation becomes:
y = 4x + 6
the area equation becomes:
y = x^2 + 3x
we graph these 2 equation plus we graph:
y = 70
y = 180
the graph of these equations looks like this:

you can see from this graph that, in order for y to be >= 180, x has to be >= 12.
you can also see from this graph that, in order for y to be <= 70, x has to be <= 16.
the equation of the area is the curved line.
the equation of the perimeter is the straight line.
the y value for the curved line has to be greater than or equal to 180.
the y value for the straight line has to be less than or equal to 70.
the intersections of the horizontal lines with the equation lines tells you what the y value of the cutover point is.
the intersections of the vertical lines with the equation lines tells you what the x value of the cutover point is.
horizontal lines are easy to draw.
vertical lines are harder.
i can get close with the vertical lines, but not right on.
in this case i did pretty good.

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the probability that any student will use a credit card because of the rewards program is .36
that is 36% divided by 100%.
the probability that a student will use a credit card for other than the rewards program is equal to 1 - .36 = .64
you ask 10 students if they use a credit card.
the probability that exactly 2 students will use a credit card because of the rewards program.
the formula for determining exactly x number of students would be:
p(x) = .36^x * .64^(10-x) * 10Cx
these are the total probabilities as far as i can see them.
p(c) is the probability of using a credit card because of the rewards program.
p(o) is the probability of using a credit card for other reasons.
x is the number of people out of 10 who are using the credit card because of the rewards program.
(10-x) is the number of people out of 10 who are using the credit card for reasons other than the rewards program.
p(x) is the probability of exactly x people out of 10 of using the credit card for the rewards program.

x	p(c)^x	        p(o)^(10-x)	10Cx	p(x)
0	1	        0.011529215	1	0.011529215
1	0.36	        0.018014399	10	0.064851835
2	0.1296	        0.028147498	45	0.164156206
3	0.046656	0.043980465	120	0.24623431
4	0.01679616	0.068719477	210	0.242386899
5	0.006046618	0.107374182	252	0.163611157
6	0.002176782	0.16777216	210	0.07669273
7	0.000783642	0.262144	120	0.024651235
8	0.000282111	0.4096	        45	0.00519987
9	0.00010156	0.64	        10	0.000649984
10	3.65616E-05	1	        1	3.65616E-05
sum of all probabilities >>>>>>>>>>>>>>>>>>>>>>	1

the answers to your questions are:

a.)         p(2)        = 0.164156206
b.)         p(>2)       = 0.759462744	
c.)         p([2,5])    = 0.816388571

p(2) is the probability of getting exactly 2 out of 10 people who are using the credit card for the rewards program.
this is just grabbing the p(2) out of the table above because each of those entries is the probability of getting exactly that number of people out of 10 who are using the credit card because of the rewards program.
p(>2) is the probability of getting more than 2 out of 10 people who are using the credit card for the rewards program.
this can be calculated by taking the sum of the probability of (3,4,5,6,7,8,9,10) people out of 10, or of taking the sum of the probability of (0,1,2) people out of 10 and subtracting that from 1.
p([2,5]) is the probability of (2,3,4,5) people out of 10 who are using the credit card because of the rewards program.
the sum of all probabilities is equal to 1 as it should be.
10Cx is the combination formula.
the total probability of exactly x people out of 10 using the credit card becsuse of the rewards program is the probability times the number of possible ways that probability can be incurred.
example:
the probability of getting exactly 1 person out of 10 would be:
.36^1 * .64^9 * 10C1 which would be equal to:
.36^1 * .64^9 * 10
this would be:
c---------
-c--------
--c-------
---c------
----c-----
-----c----
------c---
-------c--
--------c-
---------c
what this is saying is:
it could be the first person or it could be the second person or it could be the third person, etc.
the probability of each of these occurrences is .36^1 * .64^9.
since there are 10 possible ways that can happen, the total probability for 1 person out of 10 would be 10 * .36^1 * .64^9
with 2 people out of 10, it can happen 45 ways.
with 3 people out of 10, it can happen 120 ways.
etc.

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you are missing a 9 in your sequence.
the sequence should be:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
that's 10 numbers in total.
you are looking for 5 numbers in total to be equal to 1/2 of 100.
100 is an even number.
50 is an even number.
if you add 2 odd numbers together you will always get an even number.
that's why you can add 10 odd numbers together to get an even number.
when you reduce that by half, you are looking for 5 odd numbers that add up to an even number.
that can't be done.
adding an odd number of odd numbers will always result in an odd number.
you will not be able to get 5 numbers in this sequence to total up to 50 because 50 is an even number and you are wanting to add an odd number of odd numbers in order to get it.
consider this.
take your 10 numbers and pair them with each other.
(1+3) = 4
(5 + 7) = 12
(9 + 11) = 20
(13 + 15) = 28
(17 + 19) = 36
now take any of 5 of these numbers and pair them together.
(5+7) = 12
(13 + 15) = 28
the next number is all by itself and is odd.
you will never get 50 out of any 5 of these numbers.

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the cost to remove x% of the pollutant is given by the equation:
c =
if i understand this equation correctly, then the cost to remove 0% of the pollutant is equal to = 30 million dollars.
that's the fixed cost for doing nothing except setting up the necessary infrastructure in order to be able to remove the pollutant, but not actually removing any pollutant.
if i want to remove 100% of the pollution, then the cost will be = which is undefined, so i could never remove 100% of the pollution.
if i want to remove 99% of the pollution, then the cost will be which equals 3000 million dollars which is equivalent to 3 billion dollars.
based on the equation, and the fact that you are talking about x%, i believe that's the way the equation would be use.
i thought of converting % to rate by dividing it by 100, but i don't think that's proper in this case.
if i did that, then 99% would be equal to .99 and the cost would be = = 30.2999697 million.
somehow that doesn't look right.
too much fixed cost and not enough variable cost.
i would go with the original assumption.
with that assumption, the answer would be 3 billion dollars.
a graph of your equation would look like this:

the horizontal line is at 3000 million dollars which is right around the 99% point.
there's an asymptote at x = 100 because the value of y is undefined there.
looks like the cost is relatively small until you get to 90% or so.
the following graph might show those smaller numbers clearer.
the following graph shows that the cost at 90% is equal to 300 million.

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let the sides be x,y,z.
let side x be the shortest side.
y = 200 + x
z = 500 + x
the perimeter = 1900 feet.
x + y + z = 1900
substitute 200 + x for y to get:
x + 200 + x = 1900
substitute 500 + x for z to get:
x + 200 + x + 500 + x = 1900
combine like terms to get:
3x + 700 = 1900
subtract 700 from both sides of the equation to get:
3x = 1200
divide both sides by 3 to get:
x = 400
this makes:
x = 400
y = 600
z = 900
x + y + z = 1900
the answer is:
x = 400
y = 600
z = 900
x = shortest side
y = 200 + x
z = 500 + x

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x = number of units sold
cost is equal to 30 + 2.50x
sale price is 3x
c = 2.50x + 30
r = 3x
break even point is when the cost function equals the revenue function.
2.5x + 30 = 3x
subtract 2.5x from both sides of this equation to get:
3x - 2.5x = 30 which becomes:
.5x = 30
divide both sides of this equation by .5 to get:
x = 60
that's the break even point.
cost = 2.5*60 + 30 = 180
revenue = 3*60 = 180

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V = {cat, dog, bird}
W = {cat, dog, bird, mouse}
V is a subset of W because every element in V is also in W.
In fact V is a proper subset of W because every element in V is also in W and W contains other elements that are not in V.;
if W only contained {cat,dog,bird}, then:
V would be a subset of W and W would be a subset of V and both sets would be equivalent because they contained the same members.
V would not, however, be a proper subset of W.
Likewise W would not, however, be a proper subset of V.
in order for set V to be a proper subset of set W, set W would need to contain elements that are not in V.
Since it does, V is a proper subset of W.
Since V is a proper subset of W, this means that W is a proper superset of V.
If V is a subset of W, then W is a superset of V.
If W is a superset of V, then V is a subset of W.
your question asked if the sets are equal, equivalent, neither or both.
they are neither.
they are not equal and they are not equivalent.

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you want to add 3/12 and 3/4x.
you need to multiply the first fraction by 4x/4x to get:
3/12 = 12x/48x
you need to multiply the second fraction by 12/12 to get:
36/48x
now you can add them together to get:
(12x + 36)/48x
this can be reduced to:
(1x+3)/4x
the final expression is:
(x+3)/4x
the original expression is:
(3/12) + (3/4x)
take any value for x and solve it using the original expression and then solve it using the final expression and the answer should be the same.
for example:
let x = 15
the original expression yields .3
the final expression yields .3
you get the same answer which means the expressions are equivalent.
looking at what you did, i see the following:
3/12 + 3/4x = 3x/12x + 9/12x looks wrong.
in order for the denominators to be the same, they have to be a combination of 12 and 4x.
12 * 4x = 48x.
that would be a common denominator.
if you multiply the first fraction by 4x/4x, then you will get:
(3/12)( * (4x/4x) = 12x/48x
if you multiply the second fraction by 12/12, then you will get:
(12/12) * (3/4x) = (36/48x)
now you have a common denominator and you can add them together.
you get:
(12x/48x) + (36/48x) which can be combined under one denominator to become:
(12x + 36)/48x
this can be reduced by multiplying the fraction by (1/12)/1/12) to get:
(12x + 36)/48x * (1/12)/(1/12) becomes:
(x+3)/4x
your problem was in finding what the common denominator was.

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half a gallon per lawn.
5 lawns.
5 * 1/2 gallon = 2.5 gallons
3 dollars times 2.5 gallons = 7.5 dollars.

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the quadratic equation is of the form:
y = ax^2 + bx + c
a is the coefficient of the x^2 term
b is the coefficient of the x term
c is the constant
if a is positive, then the graph points down and opens up.
if a is negative, then the graph points up and opens down.
here's a graph of x^2 + 2x + 1
this one will point down and open up because the coefficient of the x^2 term is positive.

here's a graph of -x^2 + 2x + 1
this one will point up and open down because the coefficient of the x^2 term is negative.

you can find the roots of the quadratic equation by use of the quadratic formula.
that formula is:
x =
you can find the value of x for the min/max point of the quadratic equation by using the formula:
x =
you can find the value of y for the min/max point of the quadratic equation by using the formula:
y =
means that, after you found the value of x for the min/max point, you replace x in the equation with that value for x and solve for y.
an example will show you how these formulas work.
we'll work with the equation:
y = x^2 - 3x - 10
to find the roots of that equation, we set the equation equal to 0 and use the quadratic equation to find the roots.
our equation becomes:
x^2 - 3x - 10 = 0
since this is now in standard form of ax^2 + bx + c = 0, we get:
a = 1
b = -3
c = -10
we plug those values into the quadratic formula to get:
x = becomes:
x = which becomes:
x =
this leads to 2 possible values of x.
they are:
x = 5
x = 2
those are the values of x which lead to 0 for a value of y.
the roots of the quadratic equation are when the quadratic equation crosses the x-axis.
this happens when the value of y = 0.
we can find the min/max point of this equation by using the formula to get the x value.
using that formula, we get:
x = which makes x = (3/2).
the y value if found by getting which becomes which means you replace x in the equation with (3/2) and solve for y.
the equation is:
y = x^2 -3x - 10
replacing x with (3/2) gets:
y = (3/2)^2 -3*(3/2) - 10 which gets:
y = 9/4 - 9/2 - 10 which gets:
y = 9/4 - 18/4 - 40/4 which gets:
-49/4 which is equivalent to
y = -12.25
to summarize what we have:
the roots of the equation are:
x = 5
x = 2
the min/max point of the equation is:
y = -49/4 when x = 3/2
the graph of this equation is shown below:

the horizontal line is at y = -49/4.
you can see that the min/max point intersects with that line which is the y value of the min/max point.
the vertical line is at x = 3/2.
you can see that the min/max point intersects with that line which is the x value of the min/max point.
the axis of symmetry of the quadratic equation is that vertical line.
the axis of symmetry of a quadratic equation means that the part of the graph to the left of that line is an exact mirror image of the part of the graph to the right of that line.

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the argument is:
statement: If I shower in the morning, then I do not smell bad.
premise: I showered in the morning.
conclusion: Therefore, I do not smell bad.
this argument is valid.
the general form of this implied statement is:
if p then q
p
therefore q
it is one of the 4 valid forms of argument.
it would be called:
Law of Detachment (Modus Ponens)
It is a direct form of reasoning.
If p is true than q is true.
p is true, therefore q is true.
Here's the reference:
http://ux.brookdalecc.edu/fac/tlt/math/136/logic.pdf
go to page 6.

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your number are from 1 to 7.
those numbers are 1,2,3,4,5,6,7
the odd number are:
1,3,5,7
the even numbers are:
2,4,6
the probability of getting an odd number is 4/7.
the probability of getting an even number is 3/7.
you say the answer is 2/3.
since you are given that the number is even, then the possible number are 2,4,6.
if 2 out of 3 of them contain a circle, then the probability would be 2/3.
i would guess that 2 out of the 3 even numbers contain a circle.
you can send me the picture, but if my assumption is correct, you don't need to.
this would fall under the heading of p(a given b) is equal to p(a intersect b) divided by p(b).
that would be shown as:
p(a|b) = p(a^b)/p(b)
^ means intersect.
| means given.
here's how you would calculate the probability using this formula.
p(a) is the probability of getting a circle.
p(b) is the probability of getting an even number.
the formula would read:
probability of getting a circle given that the number is an even number.
suppose your data was as follows:
1 = c
2 = c
3 = c
4 = c
5 = o
6 = o
7 = o
c = circle
o = other.
you have 4 circles in the data set.
2 of them are in odd numbers
2 of them are in even numbers.
p(a) is therefore 4/7 (4 circles out of a total of 7 numbers).
p(b) is 3/7 (3 even numbers out of a total of 7 numbers).
p(a^b) would be the intersection of even numbers with circles which would be 2/7 (there are 2 even numbers that contain circles out of a total of 7 numbers).
the formula is:
p(a|b) = p(a^b)/p(b)
that comes out to be:
p(a|b) = (2/7) / (3/7)
because (a/b)/(b/c) is equal to (a/b)*(c/b), that is equivalent to:
p(a|b) = (2/7) * (7/3)
the 7 cancels out and you are left with:
p(a|b) = 2/3

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i'm not exactly sure how you got the probability of selecting a gateway computer is 4 out of 16.
since i'm a little confused about your presentation, i'll answer in general and you should be able to see how to do what you want to do from that.
suppose you have 4 computers.
let them be a,b,c,d
suppose you have 3 printers.
let them be e,f,g
suppose you have 2 monitors.
let them be h,i
since you have only 4 computers, the probability of selecting any one of them would be 1/4.
since you have only 3 printers, the probability of selecting any one of them would be 1/3.
since you have only 2 monitors, the probability of selecting any one of them would be 1/2.
the number of possible combination you have are:
4 * 3 * 2 = 24
now to your questions, using this setup rather than the setup you have.
-----
a)Determine the number of points in the sample space.
THEY SHOULD BE 24 USING MY SETUP (4 * 3 * 2)
b)Construct a tree diagram and determine the sample space.
SEE THE PICTURE AT THE BOTTOM FOR WHAT THE TREE DIAGRAM WOULD LOOK LIKE.
c)Determine the probability of selecting
c.1)a Gateway computer.
IF THE GATEWAY COMPUTER IS ONE OF 4 COMPUTERS, THEN THE PROBABILITY OF GETTING A GATEWAY COMPUTER IS 1 OUT OF 4 = 1/4.
THE ANSWER TO THIS QUESTION DOES NOT TAKE INTO CONSIDERATION THE TYPE OF PRINTER NOR THE TYPE OF MONITOR.
c.2)a Hewlett-Packard printer.
IF THE HEWLETT-PACKARD PRINTER IS ONE OUT OF 3 PRINTERS, THEN THE PROBABILITY OF GETTING A HEWLETT-PACKARD PRINTER IS 1 OUT OF 3 = 1/3.
THE ANSWER TO THIS QUESTION DOES NOT TAKE INTO CONSIDERATION THE TYPE OF COMPUTER NOT THE TYPE OF MONITOR.
c.3)a Gateway computer and a Hewlett-Packard printer.
IF THE PROBABILITY OF GETTING A GATEWAY COMPUTER IS 1 OUT OF 4 AND THE PROBABILITY OF GETTING A HEWLETT-PACKARD PRINTER IS 1 OUT OF 3, THEN THE PROBABILITY OF GETTING A GATEWAY COMPUTER WITH A HEWLETT-PACKARD PRINTER IS (1/4)*(1/3) = 1/12.
LOOKING AT THE PICTURE BELOW, YOU CAN ALSO ANSWER THE QUESTION BY COUNTING THE POINTS THAT INCLUDE THE CRITERIA YOU ARE LOOKING FOR.
THE PROBABILITY OF GETTING A GATEWAY COMPUTER, ASSUMING THAT a = GATEWAY COMPUTER, WOULD BE 6 POINTS OUT OF A TOTAL OF 24 POINTS WHICH WOULD BE EQUAL TO 1/4. THOSE POINTS WOULD BE 1,2,3,4,5,6.
THE PROBABILITY OF GETTING A HEWLETT PACKARD PRINTER, ASSUMING THAT e IS A HEWLETT PACKARD COMPUTER, WOULD BE 8 POINTS OUT OF A TOTAL OF 24 POINTS WHICH WOULD BE EQUAL TO 1/3. THOSE POINTS WOULD BE 1,2,7,8,13,14,19,20.
THE PROBABILITY OF GETTING A GATEWAY COMPUTER WITH A HEWLETT PACKARD PRINT, ASSUMING THAT a IS A GATEWAY COMPUTER, AND e IS A HEWLETT PACKARD PRINTER, WOULD BE EQUAL TO 2 POINTS OUT OF A TOTAL OF 24 POINTS WHICH WOULD BE EQUAL TO 1/12. THOSE POINTS WOULD BE 1,2.

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check out the following reference and you'll see how the method is used.
http://answers.yahoo.com/question/index?qid=20100926112059AAcVYbp
check out the following answer to another question regarding the indian method for another example of how the method is used.
http://www.algebra.com/algebra/homework/Functions/Functions.faq.question.476602.html

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here's a reference that might make some sense out of how to use this method.
http://answers.yahoo.com/question/index?qid=20100926112059AAcVYbp
the equation that you need to get the square root of should be a perfect square.
that's why you can get the square root of it fairly easily (we'll see).
i'll use that reference as a guide and see if i can solve your problem.
-----
x^2 + 12x -64 =0
a. move the constante term to the right side of the equation:
x^2 + 12x = 64
b. multiply each term in the equation for four times the coefficient of the x^2 term:
4x^2 + 48x = 256
c. square the coefficient of the original x term and add it to both side of the equation:
4x^2 + 48x + 144 = 400
d. Take the square root of both sides
this is where 4x^2 + 48x + 144 should be a perfect square.
you should get (ax + b)^2 = 4x^2 + 48x + 144
we''ll try:
(2x+12)^2 = 4x^2 + 48x + 144 = 400
I did the math and confirmed that it is true, so we are left with:
(2x+12)^2 = 400
e. Set the left side of the equation to the positive square root on the right side and solve for x
2x+12 = 20
2x = 8
x = 4
f. Set the left side of the equation equal to the negative square root on the right side and solve for x.
2x+12 = -20
2x = -32
x = -16
-----
your answers are:
x = 4 and x = -16
the graph of your original equation looks like this:

looks like we have the right answers by looking at the graph.
-----
the indian method appears to be an offshoot of the completing the squares method.
if we solved this equation by the completing the squares method, we would have done the following:
original equation:
x^2 + 12x - 64 = 0
place the constant to the right side of the equation:
x^2 + 12x = 64
take 1/2 the coefficient of the x term.
6
square it
36
add it to the right side of the equation
x^2 + 12x = 100
form your squared factor by using 1/2 the coefficient of the x term
(x+6)^2 = 100
take the square root of both sides of the equation
x+6 = +/- 10
solve for the positive square root of the right side of the equation.
x = 4
solve for the negative square root of the right side of the equation.
x = -16
you get the same answer.
with the completing the squares method, the coefficient of the x^2 term has to be equal to 1.
this is a complicating factor that makes it a little more difficult to solve.
it appears the indian method does away with that requirement.
how did they figure out to multiply by 4?
good question.
i might look into that when i have time.
not now though.
it's enough to finally figure out how to use their method.
thanks to the reference, i was finally able to.

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the statements are:
I) If the cat does not have claws, then the cat cannot scratch the furniture.
II) If the cat can scratch the furniture, then the cat has claws.
III) If the cat has claws, then the cat can scratch the furniture.
you can solve this using truth tables.
you would do so as follows:

p = cat has claws
q = cat scratches furniture
~p->~q = I
q->p = II
p->q = III
truth table:
             p   q   ~p   ~q   ~p->~q   q->p   p->q
             T   T    F    F      T       T     T
             T   F    F    T      T       T     F
             F   T    T    F      F       F     T
             F   F    T    T      T       T     T

from the truth tables, it appears that ~p->~q is equivalent to q->p because their truth tables are the same.
that means that I is equivalent to II
III is not equivalent to either one of them because its truth table is different.
if we examine the statement, we will see why this is so.
the statement are, once again:
I) If the cat does not have claws, then the cat cannot scratch the furniture.
II) If the cat can scratch the furniture, then the cat has claws.
III) If the cat has claws, then the cat can scratch the furniture.
statement II looks like it is the contrapositive of statement I.
statement I says:
if no claws, then no scratch furniture.
statement II says:
if scratch furniture, then claws.
if the statement is:
if no claws, then no scratch furniture
then the contrapositive to that statement is:
if scratch furniture, then claws.
if the statement is:
if scratch furniture, then claws
then the contrapositive to that statement is:
if no claws, then no scratch furniture.
the statement and its contrapositive are equivalent which means if the statement is true then the contrapositive is also true and vice versa.
this is the case between statements I and II.
statement III says:
if claws, then scratch furniture.
the contrapositive to this would be.
if no scratch furniture, then no claws.
all 3 statements are:
I = if no claws, then no scratch furniture
II = if scratch furniture, then claws.
III = if claws, then scratch furniture.
the contrapositive to statement III is:
if no scratch furniture, then no claws
neither I or II are the contrapositive to statement III, therefore the only equivalent statement are I and II.
It's a little convoluted, but that's the way it works out.
Use of the truth tables can make sense of the insensible, assuming you know how to use them.
p->q is only false if p is true and q is false.
q->p is only false if q is true and p is false.
~p->~q is only false if ~p is true and ~q is false.
~q->~p is only false if ~q is true and ~p is false.
that's the truth table for the implied statement.l

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a. Suppose a market research company finds that at a price of p = $20, they would sell x = 42 tiles each month. If they lower the price to p = $10, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. Hint: Write an equation using two points in the form (x,p).
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this is a little difficult to model, but it can be done as follows:
the demand equation tells you the demand for a given value of the price.
the independent variable is the price.
the dependent variable is the demand.
the facts that you are given are as follows:
if we allow p to be equal to the price and d to be equal to the demand, then:
when p = 20, d = 42
when p = 10, d = 52
the equation we will be dealing with will be a straight line equation in the form of p = mx + b.
actually, this is technically correct.
the equation we will be dealing with is:
d = mp + b
the demand is equal to m times the price + b
m is the slope
b is the d intercept which means that b is the value of d when p is equal to 0.
they actually threw you off when they gave you p = mx + b, because that would be a price equation rather than a demand equation.
anyway, our equation is:
d = mp + b
since this is a straight line equation, m is the slope and b is the d intercept.
to find the slope we need to find the change in d divided by the change in p.
you are given 2 points on the line.
they are:
when p1 = 20, d1 = 42
when p2 = 10, d2 = 52
the change in demand would be equal to d2 - d1 which would be equal to 10.
the corresponding change in price would be equal to p2 - p1 which would be equal to -10.
the slope of the line is 10/-10 which is equal to -1.
your demand equation becomes:
d = -p + b
now you need to find b which is the d intercept which is the value of d when p is equal to 0.
you find this by replacing d and p with their values for one of the points in your line.
let's take d = 42 when p = 20
d = -p + b becomes:
42 = -20 + b
add 20 to both sides of this equation and you get:
b = 62
that the value of d when p = 0.
you now have your complete equation.
it is:
d = -p + 62
if you want to graph this equation, then you need to make d = y and p = x and you get:
y = -x + 62
a graph of this equation looks like this:

now:
x is the price and y is the demand.
when the price is equal to 0, then the demand is equal to 62.
when the price is equal to 10, then the demand is equal to 52.
when the price is equal to 20, then the demand is equal to 42.
when the price is equal to 62, then the demand is equal to 0.
i drew horizontal lines at y = 42 and y = 52 to show you that the value of x will be equal to 20 when y = 42, and the value of x will be equal to 10 when y = 52.
remember that y is equal to the demand and x is equal to the price.
this occurred when we made y equal to d and x equal to p.
here's the graph again with the y = 42 and 52 lines showing. just drop a perpendicular from each intersection to find the value of x for that intersection.

it's hard to see on the graph, but you can easily solve it using the equation.
the eqution is:
y = -x + 62
when x = 10, y = 52
when x = 20, y = 42
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b. The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $300, and the supplier’s cost for a set of tile is $6 each. Let x represent the number of tile sets.
-----
not sure what you're asking here.
the equation for the total cost of a product would be the fixed cost plus the incremental cost.
the fixed cost is an amount of money that has to be paid regardless of the number of units that are produced.
the incremental cost is the additional cost per unit.
if the company allocates $300 to the fixed cost of a product, and the incremental cost for that product is $7.00 per tile, then the total cost for producing x amounts of tile would be:
y = 7x + 300
if the company builds 0 tiles, it will cost 300.
if the company builds 1 tile, it will cost 307.
if the company builds 2 tiles, it will cost 314.
etc.
Assuming that $300 is the fixed cost, then the equation is what I showed above, which is:
y = 7x + 300.
the total cost for building x products is 300 + 7*x.
-----
c. If b represents a fixed cost, what value would represent b?
not sure which b you're talking about here.
in the general slope intercept form of the equation of a straight line, the equation is:
y = mx + b
b would represent the y intercept which is the value of y when x is equal to 0.
if you are talking about the value of the fixed cost in question b, then the value of b would be equal to $300 which is the fixed cost.
it is also the value of y when x is equal to 0 in the equation of y = 7x + 300.
-----
hope this helps.
the hardest part of this was determine the demand equation. that's because the equation that was shown as an example threw me off.
here's a reference on demand questions from the tutors at algebra.com so you can see for yourself what the demand equation is all about.
http://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.187579.html

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rate * time = distance.
your rate is 70 miles per hour
your distance is 40 miles.
put that in your equation and you get:
70 * time = 40
divide both sides of that equation by 70 and you get
time = 40/70 hours.
this is equal to .571428571 hours
multiply that by 60 to get minutes.
you get:
time = (40*60)/70 which equals:
time = 34.28571429 minutes

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since 8+8 does not equal 1000, you don't have a solution.
how did you get to that point?
what is the original equation that you had to solve?

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your equation is:

this is equivalent to:

since , this becomes:
which becomes:

this is equivalent to:
in strict scientific notation where you have 1 digit to the left of the decimal point and the rest of the digits to the right of it.
since is equivalent to 61.06, then your expression became:

since , this became:
which equals:

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the reserves were 400,000 in 1980.
the reserves were 320,000 in 1990.
that means the reserves lost 80,000 barrels between 1980 and 1990.
that's 80,000 in 10 years which is an average straight line loss of 8,000 barrels per year.
your trend is a straight line.
the equation for a straight line is y = mx + b where m is the slope and b is the y intercept.
y is the number of barrels of oil reserves.
x is the number of years.
m is equal to the slope.
the slope is equal to the change in the value of y divided by the change in the value of x.
the change in the value of y is 320,000 minus 400,000 = -80,000
the change in the value of x is 1990 minus 1980 = 10
the slope is equal to -80,000/10 = -8,000.
the equation of y = mx + b becomes:
y = -8000x + b
the value of b is the value of y when x is equal to 0.
since you want 1980 to be the starting point of your equation, you subtract 1980 from all x values to get:
x = 0 means 1980
x = 10 means 1990
x = 20 means 2000
etc.
since the value of b is the value of y when x is equal to 0, then the value of b will be 400,000 because the reserves contain 400,000 barrels in 1980 which you just made equivalent to year 0.
your equation therefore becomes:
y = -8,000x + 400,000
when will the oil reserves run out?
they will run out when y = 0.
replace y with 0 in your equation and solve for x.
your equation of:
y = -8,000x + 400,000 becomes:
0 = -8,000x + 400,000
add 8000x to both sides of this equation to get:
8,000x = 400,000
divide both sides of this equation by 8000 to get:
x = 400,000/8,000 = 400/8 = 50 years
at the present rate of consumption, the oil will run out in 50 years.
a graph of the equation of :
y = -8,000x + 400,000 is shown below:

the graph shows the number of barrels in thousands.
400 is equal to 400,000
320 is equal to 320,000
etc.
the horizontal lines are at 400 and 320.
400 is the value when x = 0.
320 is the value when x = 10
that's equivalent to 400,000 when x = 0 and 320,000 when x = 10
the graph goes to 0 when x = 50.
that's when the oil runs out at the present rate of consumption.
to find the value of x when y = 320, just drop a vertical line from the intersection of the line with the equation of y = 320 (the horizontal line) and the line of the equation y = -8x + 400 (the line that slants down from the left to the right).
remember that the graph is in thousands.
the original equation is y = -8,000x + 400,000
divide the right side of the equation by 1,000 and you get the result in thousands.
the equation becomes y = -8x + 400

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you create the frequency polygon by finding the midpoint of each class and then
drawing a graph from it.
the graph would be a straight line between the midpoints on the graph.
first you have to find the midpoints.
midpoint of 0-10 = 5
midpoint of 10-20 = 15
midpoint of 20-30 = 25
midpoint of 30-40 = 35
those are the x-value of your graph.
the y-values of your graph are the frequencies in each class.
so you now have (x,y) coordinate pairs on your graph.
they are:

x       y
5       4
15      8
25      7
35      10

the following reference will show you how to draw the graph.
http://www.math-mate.com/chapter16_10.shtml

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easiest way to answer this is to graph it.

that's if you have the graphing software that makes it easy.
from the graph, it looks like this equation will be less than 0 between 4 and 5.
that stands to reason.
since this is a quadratic equation that has already been solved for the roots, it should be easy to determine what points in this equation are above 0 or below 0 or at 0.
the equation you have to work with is:
(x-5)*(x-4) = 0
by setting the expression equal to 0, you are looking for the roots.
the roots becomes:
x = 4 and x = 5
those are the zero points of the graph.
you multiply the factors together to get the original quadratic equation that gave you (x-5)*(x-4.
(x-5)*(x-4) = x^2 - 4x -5x + 20
combine like terms and you get:
x^2 - 9x + 20
if you set this equal to 0, then it becomes the standard form of the quadratic equation.
you get x^2 - 9x + 20 = 0
the standard form of the quadratic equation is ax^2 + bx + c = 0
this makes:
a = 1
b = -9
c = 20
the min/max point of the quadratic equation is given by the equation:
x = -b/2a
this becomes:
x = -(-9)/2) which becomes:
x = 4.5
That's the x value of the min/max point.
the y value of the min/max point is given by:
y = f(4.5)
you replace x with 4.5 in the equation and you get:
y = x^2 -9x + 20 which becomes:
y = (4.5)^2 -9*4.5 + 20 which becomes:
y = -.-025
your min/max point is at the (x,y) coordinates of (4.5,-.025).
now you want to determine whether this is a min point or a max point.
this is because you didn't see the graph i just showed you above.
you haven't even drawn the graph yet.
you're working just from the equation.
you look at the exponent of the x^2 term.
if it is positive then the graph points down and opens up.
if it is negative then the graph points up and opens down.
if the graph points down, then the min/max point is a min point.
if the graph points up, then the min/max point is a max point.
the graph is pointing down because the coefficient of the x^2 term is positive.
that would be the a in ax^2 which was equal to 1.
so your graph is point down.
the roots of the eqution are at x = 4 and x = 5
the min point is negative.
it will stay negative between the roots and it will be positive outside of the roots.
your equation is therefore >= 0 when:
x < 4 or x > 5

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let x = number of questions that are 2 points each.
let y = number of questions that are 3 points each.
since the total number of questions is 35, then your first equation is:
x + y = 35
since each x gets you 2 points and each y gets you 3 points, then your ssecond equation is:
2x + 3y = 100
solve these 2 equations simultaneously for your answer.
use second equation to solve for y in terms of x.
you get y = 35 - x
substitute for y in the first equation to get:
2x + 3(35-x) = 100 which becomes:
2x + 105 - 3x = 100 which becomes:
-x + 105 = 100
add x to both sides of this equation and subtract 100 from both sides of this equation to get:
x = 5
since x + y = 35, this means that y = 30
you have 5 questions that are 2 points each and 30 questions that are 3 points each for a total of 5 + 30 = 35 questions with a total of 10 + 90 = 100 points.

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you're on the right track, but you needed to do some other type thinking before getting to the solution, which is a rule that i will explain to you after.
the equation is:
nC12 = nC8
this formula translates to:
n!/(12!*(n-12)!) = n!/(8!*(n-8)!)
cross multiply to get:
n! * (8!*(n-8)!) = n! * (12!*(n-12)!)
divide both sides of this equation by n! to get:
8! * (n-8)! = 12! * (n-12)!
divide both sides of this equation by 8! and divide both sides of this equation by (n-8)! to get:
(n-8)! / (n-12)! = 12!/8!
12! is equal to 12 * 11 * 10 * 9
(n-8)! / (-12)! is equal to (n-8)*(n-9)*(n-10)*(n-11)*(n-12)! / (n-12)! which is equal to:
(n-8)*(n-9)*(n-10)*(n-11)
our equation becomes:
(n-8)*(n-9)*(n-10)*(n-11) = 12*11*10*9
the left side of this equation will be equal to the right side of this equation if:
n-8 = 12
n-9 = 11
n-10 = 10
n-11 = 9
all of these equation would be true if n = 20.
that's your answer.
n = 20.
20C12 = 125970
20C8 = 125970
they're the same.
note that 12 + 8 = 20
that's the rule.
you can use it with any problems of this type to immediately get the answer.
an example:
nC13 = nC22
13 + 22 = 35
your answer is n = 35
35C13 = 1476337800
35C22 = 1476337800
The basic properties of combinations allows this to happen.
that property is that nCx = nC(n-x)
from that formula, you get x + (n-x) = n
a simpler example:
take n = 7
your values for each of the possible combinations are:
7C0 = 1
7C1 = 7
7C2 = 21
7C3 = 35
7C4 = 35
7C5 = 21
7C6 = 7
7C7 = 1
you can see that 7Cx = 7C(7-x)
this makes 7C0 = 7C7
this makes 7C1 = 7C6
this makes 7C2 = 7C5
etc.
Add the x and the (n-x) together and you get 7 every time.
your problem as nC12 = nC8
add 12 and 8 together and you get 20.

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i believe you are looking for the 5th root of (-1/32)
that would be written as:
root(5,(-1/32)) in algebra.com language.
put 3 { brackets in front of it and 3 } brackets behind it and you get:

since 32 is equal to 2^5, this would be written as:

since the root of (a/b) is equal to (the root of a) / (the root of b), then this expression can be re-written as:

since the fifth root of (-1) is equal to -1 because (-1)^5 = -1, and:
since the fifth root of 32 is equal to 2 because (2)^5 = 32, then:
your expression is equivalent to:
(-1/2).
you may confirm this is true by using your calculator to find the fifth root of (-1/32).
the calculator should tell you that it's equal to -.5.

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plot the points on your graph.
draw the figure.
find the diagonal.
get the endpoints of the diagonal.
solve for the length of the diagonal using the length formula.
your 4 points are:

A         (-2,3)
B         (-2,9)
C         ( 5,9)
D         ( 5,3)

Looks like A will be directly below B because they both have the same x coordinate.
Looks like C will be directly to the right of A because they both have the same y coordinate.
Looks like D will be directly below C because they both have the same x coordinate.
Looks like D will be directly to the right of A because they both have the same y coordinate.
I would guess you have a rectangle.
It will have 2 diagonals.
They will be from A to C and from B to D.
Since this is a rectangle, they should both have the same length because the diagonals of a rectangle are equal in length to each other.
First we'll solve for the length of AC and then we'll solve for the length of BD.
AC extends from point A to point C.
Those points are:

point        x value        y value
A               -2             3
C                5             9

BD extends from point B to point D.
Those points are:

point        x value         y value
B              -2               9
D               5               3

The formula for length of a line is:
length of a line equals
Substitute the points for A and C in the formula and you will get a length of for AC.
Substitute the points for B and D in the formula and you will get a length of for BD.
Their length are the same, as they should be if we are looking at a rectangle.
Based on the points, I believe we are.
A picture of what this object would look like is shown below.

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if the water is filtered, then it does not contain lead.
the water does not contain lead.
therefore, the water is filtered.
-----
this argument is not valid.
if is not valid by the fallacy of the .....
here's a reference that contains all the valid argument types and the fallacy types.
http://ux.brookdalecc.edu/fac/tlt/math/136/logic.pdf
your answer is on page 7 of this document.
it's called "fallacy of the converse"
the symbolic form would be:

premise 1:        p->q
premise 2:        q
conclusion:       therefore p

to understand why this is a fallacy, you need to go to the truth tables for the implies statement.
that truth table is shown below:

p q p->q
T T T
T F F
F T T
F F T

you can see from this table that the only time q if false is when p is true and q is false.
if p is false, then q can be true or false and the statment is still true.
the fallacy of the converse states that the following logic is invalid.
p->q
q
therefore p
it is saying that because q is true, then p must be true.
but you can see from the truth table for the implied statement that when q is true, p can either be false or true, therefore, you can't make that statement shown in the fallacy of the converse.
just because p implies q doesn't mean that q implies p.
the converse of a statement is shown below in symbolic form.

statement:    p->q
converse:     q->p

this can be true some of the time and can be false some of the time. it is not always true.
an example of when it is not true would be:

statement:     all cats are mammals
converse:      all mammals are cats

while the statement is true, its converse is clearly not true.
that's why your problem statement is invalid by the fallacy of the converse.
your statement:

statement:     if the water is filtered, it does not contain lead.
converse:      if the water does not contain lead, it is filtered.

nowhere in the statement does it implied the converse is true.
if the water does not contain lead, it may or may not be filtered.
not all water contains lead.