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i appears there's a vertical asymptote at x = -1.
the following table helps show that.

x	y = log(2,x+1)
-2	#NUM!
-1	#NUM!
-0.9999	-16.60964047
-0.9	-3.321928095
-0.8	-2.321928095
-0.7	-1.736965594
-0.6	-1.321928095
-0.5	-1
0	0
1	1
2	1.584962501
3	2
4	2.321928095
5	2.584962501
6	2.807354922
7	3
8	3.169925001
9	3.321928095

you cannot take a log of a negative number of or 0.
that is why the first 2 values in the table are invalid.
that is why the graph stops as just to the right of x -1
log (x+1) will be log(0) when x is equal to -1.

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3.2 * 460 = 1472 which means the plan has traveled 1472 miles in 3.2 hours at 460 mph.
that part is correct.
the rest is trigonometry.
make a graph and then make a line that makes an angle of 75 degrees with the x-axis.
the line starts at the origin of the graph and extends upward to the right at the 75 degree angle from the x-axis.
you now can make a right triangle.
the hypotenuse is 1472 miles long.
the horizontal leg of this triangle represents the number of miles he travels in an easterly direction and the vertical leg of this triangle represents the number of miles he travels in a northerly direction.
the horizontal distance he travels we'll call x
the vertical distance he travels we'll call y
the hypotenuse we'll call z.
since this is a right triangle, then we get the following trigonometric relationships.
sin(75) = y/1472
cos(75) = x/1472
we use these equations to solve for x and y
y = 1472 * sin(75)
x = 1472 * cos(75)
use your calculator to solve for x and y
you should get:
y = 1421.842816
x = 380.9816344
x is the number of miles he traveled in an easterly direction.
y is the number of miles he traveled in a northerly direction.
z is the number of miles he traveled in a combined easterly and northerly direction at a 75 degree angle from the east.
the following diagram shows you what i mean.

in the diagram, the numbers are rounded.
z = 1472
x = 381
y = 1422
unfortunately, 1422 looks a lot like 1472 but if you look closely there is a difference. just bad handwriting on my part - sorry for that.

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he spent 1/6 of his life as a child, 1/12 as an adolescent, and 1/7 as a bachelor.
5 years after he's married, he has a son.
son dies 4 years before his father at half his father's final age.
how long did he live?
if we add up 1/6 and 1/12 and 1/7, we get 33/84.
this should mean that he dies at 84 and has 51 years left to live.
if that holds up, then we found the answer.
let's see if it does.
assuming we're right, then he gets married at 33 years of age.
5 years later he has a son when he is 38 years old.
based on our assumption, he dies at 84.
his son has died 4 years earlier at half his age when he dies which means his son has died at age 42.
here's the timeline.

his age          his son's age        description of event
38                     0                 his son is born.
80                     42                his son dies.
84                                       he dies.

when his son dies at 42, he is 80 (80 - 38 = 42)
4 years later he dies at 84.
numbers appear to check out so the solution is good.
it appears the only information we needed was to be able to add up 1/6 + 1/12 and 1/7 to find out how much of his life was consumed when he got married.
the remainder from the denominator is what he had left.
the denominator is how long he lived.

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this time around i'll take you through this the way i would look at it.
the equation is 144 - 25t + t^2
first thing i would do is re-order the terms so the highest level term is on the left and the lowest level term is on the right.
the equation becomes t^2 + 25t - 144
next thing i would do is notice that the coefficient of the t^2 term is equal to 1.
this is good because it makes trying to find the factors of this equation easier.
next thing i would do is determine what factors i need to multiply together to get -144.
all of the possible combinations i could find are:
-1 * 144 = -144
-2 * 72 = -144
-3 * 48 = -144
-4 * 36 = -144
-6 * 24 = -144
-8 * 18 = -144
-9 * 16 = -144
-12 * 12 = -144
-16 * 9 = -144
-18 * 8 = -144
-24 * 6 = -144
-36 * 4 = -144
-48 * 3 = -144
-72 * 2 = -144
-144 * 1 = -144
these are all of the possible combinationss of factors that, when you multiply them together, you get -144.
now i have to find which of these can be added together to get +25.
here goes:
-1 + 144 = 143
-2 + 72 = 70
-3 + 48 = 45
-4 + 36 = 32
-6 + 24 = 18
-8 + 18 = 10
-9 + 16 = 7
-12 + 12 = 0
-16 + 9 = -7
-18 + 8 = -10
-24 + 6 = -18
-36 + 4 = -32
-48 + 3 = -45
-72 + 2 = -70
-144 + 1 = -143
none of these add up to +25.
therefore, there are no factors that are integers and the polynomial is prime.

as i told you before, you can graph the equation to see if it has any roots and where those roots are and how many you have (you can have 2 or 1 or none).

you can always solve a quadratic equation using the quadratic formula. it will also tell you if the roots are integers and if they are real and how many of them you have (you can have 2 or 1 or none)

this is the process that is followed when you are looking for factors to the equation.

here's an example where the process is used and the factors can be found.

suppose the equation is x^2 + 5x + 4.

you want to find the roots.

take the constant term and determine what factors can be multiplied together to equal it.

those factors are:
4 * 1 = 4
2 * 2 = 4
1 * 4 = 4
-4 * -1 = 4
-2 * -2 = 4
-1 * -4 = 4

now look for which of these factors will add up to equal the coefficient of the x term.

you get:
4 + 1 = 5 *****
2 + 2 = 4
1 + 4 = 5 *****
-4 + -1 = -5
-2 + -2 = -4
-1 + -4 = -5

you can see that 4 + 1 or 1 + 4 will add up to 5 (the coefficient of the middle term) and that 4 * 1 or 1 * 4 will multiply together to equal 4 (the constant term)

those are your potential factors.

your equation is x^2 + 5x + 4
try (x + 4) * (x + 1)
when you multiply these factors together you get:
x^2 + x + 4x + 4 which simplifies to x^2 + 5x + 4
bingo !!!!!
you found the factors.

they are (x + 4) and (x + 1)
now you want to find the roots.
you set (x + 4) * (x + 1) = 0
you set each of the factors equal to 0
you get:
x + 4 = 0 and x + 1 = 0
you solve for x in each equation and you get:
x = -4 and x = -1
those are the roots of your equation.

the same method was used only this time you were able to find the factors.

you can graph both equation we worked on.

the first equation is t^2 + 25t - 144
in order to graph it, you have to replace t with x to make the equation:
x^2 + 25x - 144.
now you can use the graphing software to graph it.
the graph will look like this:

you can see from the graph that the roots do not appear to be integers which was confirmed by the fact that you couldn't find the factors.

the second equation is x^2 + 5x + 4
the graph will look like this:

this graph looks more like the roots are integers which means you should be able to factor it which we confirmed by finding the factors.

if in doubt, use the quadratic formula.
that will find you the roots, tell you whether you have roots, tell you whether those roots are integers or now and tell you how many roots you have (0, 1, or 2).
if the quadratic formula gives you integers as roots, then you can factor the equation.

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y = -2x + 8
y is the height of the candle in inches.
x is the amount of time it burns in hours.
in 4 hours the candle should burn down to nothing.
domain of the function is x = 0 to 4.
range of the function is y = f(x) = 0 to 8
graph of the function looks like this:

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triangle is DEF
Angle D and F are congruent.
this means that sides opposite those angles are also congruent.
that would be sides DE and EF
the length of side DE is 10x^2 - 100
the length of side EF is 5x^2 + 40x
set these sides equal to each other and you get:
10x^2 - 100 = 5x^2 + 40x
subtract 5x^2 and 40x from both sides of the equation and you get:
10x^2 - 100 - 5x^2 - 40x = 0
combine like terms and you get:
5x^2 - 40x - 100 = 0
factor out a 5 and the equation becomes:
x^2 - 8x - 20 = 0
factor this to get:
(x-10) * (x+2) = 0
solve for x to get x = 10 or x = -2.
since x can't be negative, then x = 10 is the solution you are looking for.
now that you know that x = 10, you can solve for the perimeter by replacing x with 10 in each equation and then adding up the result.
your equations are:
10x^2 - 100
5x^2 + 40x
15x
when x = 10, these equations become
10*10^2 - 100 = 1000 - 100 = 900
5*10^2 + 40*10 = 5*100 + 400 = 500 + 400 = 900
15*10 = 150
the sum of all of these is 1800 + 150 which gets you a perimeter of 1950.

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you can solve this by graphing the equation and you can then visually see the regions that the solution lies in.
the graph looks like this.

it confirms the algebraic solution provided by edwin.
the values between -1 and 1 are not mathematically possible and therefore don't show up on the graph.

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focus is 0,2 and the directrix is y = -4
the distance between the focus and the directrix is equal to (0,2) to (0,-4) which is equal to 6
that makes 2p = 6 which makes p = 3
there are 3 possible equations for this parabola.
they are:
ax^2 + bx + c = 0 (standard form)
y = a(x-h)^2 + k (vertex form)
y = (x-h)^2/4p + k (conics form)
another form of the equation in the conics form is:
4p(y-k) = (x-h)^2
we should be able to work either either one because they are equivalent.
we should be able to solve for all of them.
let's see if we can.
since p = 3, then 4p in the conics form must be equal to 12.
since a = 1/4p, this mean that a in the vertex form must be equal to 1/12.
(h,k) is the vertex of the parabola.
since you know the focus, then you can solve for the vertex because the vertex is halfway between the focus and the directrix.
you're going from (0,2) to (0,-4) which gets you to a vertex of (0,-1) because that is halfway between (0-2) and (0,-4).
with a vertex of (0,-1), we get h = 0 and k = -1.
we may be able to piece together some of the equations now.
we have:
h = 0
k = -1
4p = 12 in the conics form
a = 1/12 in the vertex form
note that a in the vertex form is not the same as a in the standard form.
we'll start with the conics form.
the equation is:
y = (x-h)^2/4p + k (conics form)
substitution gets us:
y = x^2/12 - 1
let's try the vertex form next.
the equation is:
y = a(x-h)^2 + k (vertex form)
substitution gets us:
y = 1/12 * x^2 - 1 which is equivalent to:
y = x^2/12 - 1
this is the same as the conics form so we're doing good so far.
let's try the standard form next.
the equation is:
ax^2 + bx + c = 0 (standard form)
since we know that the equation is y = x^2/12 - 1, we just set y = 0 and we get:
x^2/12 - 1 = 0
in standard form, this means that:
a = 1/12, b = 0, and c = -1
it's not clear to me if the a in the standard form is the same as the a in the vertex. it does seem to work out in this example so maybe it is.
a graph of this equation of y = x^1/12 - 1 is shown below:

from the graph, you can see that the focus is at (0,2) and the vertex is at (0,-1) and the point on the line y = -4 that you would measure to is the point (0,-4) which is in line with the focus and the vertex and lies on the axis of symmetry which is x = 0.
the reference that i used to answer this question is at:
http://www.purplemath.com/modules/parabola.htm
this is an excellent reference and well worth reading to understand this better.
it's a little difficult to follow but just be careful of all the a's and b's and c's and p's and you should be able to figure it out.
be careful though because they talk about the standard equations that have the parabola facing up or down, and the sideways equations that have the parabola facing left or right.
ideally, you should be understand both, but for this problem you only need to understand the up and down equations.
those are the ones i showed you above.
i got them from this reference so they should track with what the reference is saying.
the reference is 4 pages long. what you see when you click on the link is the first page only.
good luck.

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direct proportion is z = kx
inverse proportion is z = k/y
direct and inverse proportion together is z = kx/y
z = volume of gas in cubic meters
x = temperature in kelvin
y = pressure in kilograms per square meter.
when volume of gas is 20 cubic meters and temperature is 100 kelvin and pressure is 15 kilograms per square meter, equation of z = kx/y becomes 20 = 100k/15.
solve for k to get k = 20*15/100 which results in k = 3.
now that you know k, you can use that to solve the problem.
when the temperature is 150 kelvin and the pressure is 20 kilograms per square and k = 3, the equation of z = kx/y becomes z = 3*150/20 which becomes z = 22.5 cubic meters.
that's your answer.

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the answer is:
5/18, 7/18, 1/2, 11/18
your equation looks like it is (1/6 + (1/9)*n
n goes from 1 to ........
the first term has n = 1
the second term has n = 2
etc.
when n = 1, the equation is 1/6 + 1/9 which results in 5/18
when n = 2, the equation is 1/6 + 2/9 which results in 7/18
when n = 3, the equation is 1/6 + 3/9 which results in 9/18 which results in 1/2
when n = 4, the equation is 1/6 + 4/9 which results in 11/18
if you look at your selections, the 4th selections appears to be it.
that reads:
5/18 followed by 7/18 followed by 1/2 followed by 11/18.

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let x = munber of hamburgers
let y = number of medium cokes.
your equations become:
2x + y = 770
3x + 2y = 1260
solve for y in the first equation to get y = 770 - 2x
substitute for y in the second equation to get:
3x + 2*(770 - 2x) = 1260
simplify to get:
3x + 1540 - 4x = 1260
combine like terms and subtract 1540 from both sides of the equation go get:
-4x = 1260 - 1540 which becomes:
-x = -280
multiply both sides by -1 to get:
x = 280
the number of calories in a mcdonald's hamburger is 280.
replace x with 280 in either equation and solve for y.
we'll use the equation of:
2x + y = 770
replace x with 280 and you get:
560 + y = 770
subtract 560 from both sides to get:
y = 770 - 560 which becomes:
y = 210
the number of calories in a medium code is 210.
you get:
x = 280
y = 210
x = number of calories in a hamburger.
y = number of calories in a medium coke.
2 hamburgers and 1 medium coke is 2 * 280 + 1 * 210 which is equal to 770 calories.
3 hamburgers and 2 medium cokes is 3 * 280 + 2 * 210 which is equal to 840 + 420 which is equal to 1260 calories.
numbers check out so the answer is good.

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there are 1,000 meters in a kilometer so Jerome rides his bike 10,000 meters every day.

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if you don't know how to do it, then read this tutorial.
http://www.purplemath.com/modules/synthdiv.htm
it will explain how it's done.
if you follow their guidelines, you will get something that looks like this:

your divisor of x + 5 becomes the multiplier of -5 because when you set x + 5 = 0 and solve for x, you get x = -5.
you bring down the coefficients of all the terms.
you then bring down the first coefficient of 3.
you then multiply 3 by -5 to get -15 and add that to 11 to get - 4
you then multiply -4 by -5 to get + 20 and add that to - 20 to get 0
you then multiply 0 by -5 to get + 0 and add that to + 7 to get + 7
you then multiply + 7 by - 5 to get - 35 and add that to + 35 to get 0
your anser is 3x^3 - 4x^2 + 7 with a remainder of 0.
check out the picture and look at the tutorial and you should be able to figure out how it was done.

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good question.
what is 346 in base 8 minus 405 in base 8.
assuming these numbers are already in base 8, then you would subtract 346 from 405 and then give the answer the negative sign because the number you are subtracting is larger than the number you are subtracting from.
you would get:
346 - 405 is equal to - (405 - 346)
you would try to subtract the 6 from the 5 which you can't do because the 6 is larger than the 5.
you need to borrow from the next position.
since the next digit over is 0, you need to go over 1 more digit and borrow from the 4.
the 4 becomes 3.
the 0 becomes 8 (remember you're in base of 8 not base of 10).
now you can borrow from the second digit, so the second digit becomes 7 and the first digit becomes 5 + 8 = 13
you can now subtract 6 from 13 to get 7 for the first digit
you can now subtract 4 from 7 to get 3 for the second digit.
you can now subtract 3 from 3 to get 0 for the first digit.
the answer should be -37 in the base of 8.

a good way to check is to convert everything to base of 10, perform the operation, and then convert everything back to the base of 8.

to convert 405 from the base of 8 to the base of 10, do the following:
4 * 8^2 + 0 * 8^1 + 5 * 8^0 = 261
405 in the base of 8 is equivalent to 261 in the base of 10.

to convert 346 from the base of 8 to the base of 10, do the following:
3 * 8^2 + 4 * 8^1 + 6 * 8^0 = 192 + 32 + 6 = 230.
346 in the base of 8 is equivalent to 230 in the base of 10.

now subtract 261 from 230 to get an answer of -31 in the base of 10.

now convert -31 from the base of 10 to the base of 8.

the way to convert from the base of 10 to the base of 8 is to repeatedly divide the number by 8 and then use the remainders in reverse order from how they were created.

as an example, since we already know that 405 in the base of 8 is equal to 261 in the base of 10, we will convert 261 from the base of 10 to the equivalent number in the base of 8.
if we get 405 then we know we did it right.

261 divided by 8 = 32 with a remainder of 5
32 / 8 = 4 with a remainder of 0
4 / 8 = 0 with a remainder of 4
going with the remainders in reverse order of ho they were created, you get 405 which is the the number 261 in base 8 format.

i'll convert 230 in the base of 10 to its equivalent number in the base of 8.
if i get 346, then i did it right.

230 / 8 = 28 with a remainder of 6
28 / 8 = 3 with a remainder of 4
3 / 8 = 0 with a remainder of 3
gathering my remainders in the reverse order from which they created, i get 346 as the equivalent number in the base of 8.
since this agrees with what i already knew, then i did it correctly.

now back to the original problem.
in base of 10, the difference was -31

convert this to octal format as follows:
31 / 8 = 3 with a remainder of 7
3 / 8 = 0 with a remainder of 3
the number in octal format is -37.

this agrees with the arithmetic up front so we're good.
i got the same answer both ways.
way 1 was performing the arithmetic directly in the base of 8.
way 2 was converting everything to the base of 10 and performing the arithmetic and then converting the answer back to the base of 8.

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here's the deal.
you have 11 staff members
there is 1 professor among the group.
the fact that 7 of the staff members are women is irrelevant to the problem therefore this is extraneous information that you can ignore.
i'll solve it 2 ways.
both ways will get the same answer which confirms that the answer has a high probability of being right.
combination formula used is C(n,x) where n is the total possible choices you have and x is the total possible choices that you want.
the formula for C(n,x) is n! divided by (x! * (n-x)!).
if you have a good scientific calculator that does combination formulas for you, you can also use that.

anyway here goes with method 1.
the probability that you will get the professor for one of the 6 questions is equal to 1 minus the probability that you will not get the professor for one of the 6 questions.
that probability is equal to 10/11 * 9/10 * 8/9 * 7/8 * 6/7 * 5/6 which is equal to (10! / 4!) / (11! / 5!) which is equal to .45454545.....
1 minus .45454545..... is equal to .54545454.....
that's the probability using method 1.

method 2 is done as follows:
this method uses the combination formula.
the solution is the number of ways you can get the professor divided by the number of ways you can get anybody.
the number of ways you can get anybody is equal to C(11,6) which is the number of ways you can get any of the 11 staff members to be on a team of 6.
C(11,6) is equal to 11! / (6! * 5!) which is equal to 462.
you need one professor and you have 1 to choose from so the number of ways you can get the professor is 1C1 = 1.
you need 5 other members on the team of 6 that are not professors and you have 10 to choose from so the number of ways you can get the additional 5 members that are not professors is C(10,5) = 252.
the total number of ways you can get the professor and the non-professors to be on the team of 6 is therefore equal to 1 * 252 = 252
the number of ways of getting the professor divided by the number of ways of getting anybody is therefore equal to 252/462 = .54545454....

obviously method 1 was easier in this problem.
that may not be as true in other problems where you can't just get 1 minus the probability of not getting the professor.
both methods match so either method will get you the solution.
the second method is trickier but instructive because sometime you are taught to do it this way and it helps to understand how to do it.

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L = length
W = width
length is 7 feet less than twice the width
L = 2W - 7
the brick border around the garden adds 10 to the length and 10 to the width (2 times 5 for each 1 because the border is on both side of the length and on both sides of the width).
the area with the border is equal to 240.
area = (L+10) * (W + 10) = 240
since L = 2W - 7, replace L with 2W - 7 to get:
area = (2W - 7 + 10) * (W + 10) = 240
simplify to get:
(2W + 3) * (W + 10) = 240
simplify by performing the indicated operations to get:
2W^2 + 23W + 30 = 240
subtract 240 from both sides to get:
2W^2 + 23W - 210 = 0
factor to get:
(2W + 35) * (W - 6) = 0
solve for W to get:
W = -35/2
W = 6
W can't be negative so you get W = 6 as the solution to the quadratic equation.
since L = 2W - 7, this means that:
L = 2(6) - 7 = 5
you have:
L = 5
W = 6
add 10 to both of these to include the border to get:
L + 10 = 15
W + 10 = 16
area = (L + 10) * (W + 10) = 15 * 16 = 240
area checks out ok.
L = 2W - 7 becomes 5 = 2(6) - 7 which becomes 12 - 7 which becomes 5 checks out as well since L does equal 5.
answer is confirmed as good.
answer is:
L = 5
W = 6
those are the dimensions of the garden without the brick border.
area of the garden without the brick border is equal to 5*7 = 35 square feet.
you weren't asked that, however, so you would not include it as part of your answer.

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log.a means log to the base of a (my interpretation not generally used).
your equation becomes:
log.a(x) / log.a/b(x) = 1 + log.a(1/b)
you want to prove this identity is true.
you need to use the log base conversion formula of log.d(x) = log.e(x) / log.e(d).
you use this conversion formula on log.a/b(x) to get:
log.a/b(x) = log.a(x) / log.a(a/b)
your equation of:
log.a(x) / log.a/b(x) = 1 + log.a(1/b) becomes:
log.a(x) / (log.a(x) / log.a(a/b)) = 1 + log.a(1/b)
since h/(i/j) is equivalent to h*(j/i), your equation becomes:
log.a(x) * log.a(a/b) / log.a(x) = 1 + log.a(1/b)
since log.a(x) / log.a(x) cancels out, your equation becomes:
log.a(a/b) = 1 + log.a(1/b)
since log(m/n) = log(m) - log(n), your equation becomes:
log.a(a) - log.a(b) = 1 + log.a(1) - log.a(b)
if you add log.a(b) to both sides of the equation, it will cancel out and you'll get:
log.a(a) = 1 + log.a(1)
since log.a(1) = 0, your equation becomes:
log.a(a) = 1
since log.a(a) = 1, your equation becomes:
1 = 1 which is true confirming the identity as being valid.
log.a(1) is equal to 0 based on the following logic.
log.a(1) = y if and only if a^y = 1
a^y = 1 if and only if y = 0
this makes log.a(1) = 0
log.a(a) is equal to 1 based on the following logic.
log.a(a) = y if and only if a^y = a
a^y = a if and only if y = 1
this makes log.a(a) = 1
the key to solving this is the use of the log base conversion formula which is.
log.k(x) = log.p(x) / log.p(k)
as an example, take log.2(8) = 3
this is true because 2^3 = 8
convert this to log.10
log.2(8) = log.10(8) / log.10(2)
use your calculator to see that the answer is going to be 3.

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since 17 is a prime number, one of the numbers has to be 1 and the other number has to be 17.

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the 3d length has to be less than the other 2.
this means that the maximum length of the 3d piece has to less than 8/3 feet.
if we're assuming that there has to be something left over, then the minimum length of the 3d piece has to be greater than 0.
so we have a minimum length of the 3d piece of greater than 0 and a maximum length of the 3d piece of smaller than 8/3.
anything in between should be ok.
example:
let the 3d piece be equal to 1 foot.
this means the 2 other pieces are 7/2 feet each.
let the 3d piece be equal to 2 feet.
this means the 2 other pieces at 6/2 feet each.
as long as you don't need to have integers, there are an infinite number of lengths that would satisfy the requirements.
if you assume some discrete amount, like no length could have an measurement that doesn't end in an inch, then you now have a finite number of possible solutions.
in fact, this probably means that the 3d piece needs to be in 2 inch increments which will allow the 2 equal pieces to be in 1 inch increments.
example:
3d piece equals 1 foot which means the 2 equal pieces equal 7/2 = 3.5 feet each.
3.5 is equivalent to 3 and 6 inches which is in increments of an inch so we're good.
if we increase 3d section to 1foot 1 inch then we have 6 feet 11 inches left which we divide by 2 to get 3 feet 5.5 inches which is less than inch increments so we're not good.
we would need to increase 3d section to 1 foot 2 inches (increments of 2 inches) to get 1 inch increments on the equal segments.
1 foot 2 inches from 8 feet leaves 6 feet 10 inches which divided by 2 equals 3 feet 5 inches for each of the equal segments.
not exactly sure what you were looking for but this is the solution based on my understanding of what you are asking.

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i believe you are looking for the natural log of root4(9r^8s)
root4 is equivalent to taking the fourth root of which is equivalent to raising to the 1/4 power, so we get:
ln ((9r^8s)^(1/4))
this becomes:
(1/4) * ln ((9r^8s) which becomes:
(1/4) * [ ln(9) + ln(r^8s) ] which becomes:
(1/4) * ln(9) + (1/4) * ln(r^8s) which becomes:
(1/4) * ln(9) + (1/4) * 8 * s * ln(r)
to confirm this was done correctly, i let r = 2 and s = 3 and i solved using both the original expression and the final expression and got the same answer of 4.70819 rounded to 5 decimal places.
the concepts you use are:
log(a^b) = b*log(a)
log((a^b)^c) = c*log(a^b) = b*c*log(a)
log(a*b) = log(a) + log(b)

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the answer is x = 4
here's how.
your equation is:
log4(4x) + 2log4(x) = 4
since a*log(b) = log(b^a), this equation becomes:
log4(4x) + log4(x^2) = 4
since log(a) + log(b) = log(a*b), this equation becomes:
log4(4x * x^2) = 4
simplify this to get:
log4(4x^3) = 4
since logb(x) = y if and only if b^y = x, this equation becomes:
4^4 = 4x^3
since 4^4 = 256, this equation becomes:
256 = 4x^3
divide both sides of this equation by 4 to get:
64 = x^3
take the third root of both sides of this equation to get:
x = 4

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44 coins in total.
n = number of nickels.
d = number of dimes
q = number of quarters.
there are twice as many dimes as nickels.
d = 2n
8 fewer nickels than quarters.
n = q - 8
how much money is in the bank.
the amount of money in the bank is determined as follows:
.05n = amount of money in nickels.
.1d = amount of money in dimes.
.25q = amount of money in quarters.
you have 2 equations.
n + d + q = 44
.05n + .1d + .25q = amount of money in the bank.
you need to solve for n, d, and q and then you can determine how much money is in the bank.
start with the equation of n + d + q = 44
you are given that d = 2n so you can substitute 2n for d which makes the equation becomes:
n + 2n + q = 44
you are given that n = q - 8 so you can substitute q - 8 for n which makes the equation become:
q - 8 + 2(q - 8) + q = 44
since you now have 1 unknown in 1 equation, you can solve for q.
simplify the equation to get:
q - 8 + 2q - 16 + q = 44
combine like terms to get:
4q - 24 = 44
add 24 to both sides of the equation to get:
4q = 44 + 24 = 68
divide both sides of the equation by 4 to get:
q = 68/4 = 17
since you know that n = q - 8, this means that n = 9
you have:
q = 17
n = 9
since you know that d = 2n, this means that d = 18
now you have:
q = 17
n = 9
d = 18
the first equation you had to solve is n + d + q = 44
replace q with 17 and n with 9 and d with 18 and you get:
9 + 18 + 17 = 44 which becomes 44 = 44 which is true so the numbers for n and d and q look good.
now to the money.
the equation is:
.05n + .1d + .25q = the money
replace n with 9 and d with 18 and q with 17 and you get:
.05(9) + .1(18) + .25(17) = .45 + 1.8 + 4.25 which is equal to 6.5 dollars that you have in the bank.

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basically you're talking about the intersection of 2 lines whose equations are:
4x + 3y - 13 = 0
2x - y - 4 = 0
the fact that these lines are altitudes to sides of a triangle doesn't change the fact that you are simply looking for the point of their intersection.
if there was additional information they were looking for, it would make a difference, but the way the problem is structured, all that other information is extraneous to the heart of the problem which is to find the intersection of the 2 lines.
the 2 equations are, once again:
4x + 3y - 13 = 0
2x - y - 4 = 0
you need to solve these equations simultaneously to get the solution.
you need to first transform them to standard form which is ax + by = c
4x + 3y - 13 = 0 becomes 4x + 3y = 13 once you convert it.
2x - y - 4 = 0 becomes 2x - y = 4 once you convert it.
your 2 equations are now:
4x + 3y = 13
2x - y = 4
we will solve by elimination.
multiply the second equation by 2 to get:
4x - 2y = 8
your 2 equations are now:
4x + 3y = 13
4x - 2y = 8
subtract the second equation from the first equation to get:
5y = 5
divide both sides of this equation by 5 to get:
y = 1
substitute for y in either original equation to find the value of x.
we'll use:
4x + 3y = 13
replace y with 1 to get:
4x + 3 = 13
subtract 3 from both sides of this equation to get:
4x = 10
divide both sides of this equation by 4 to get:
x = 2.5
your solution should be:
x = 2.5
y = 1
this solution should apply to both equation.
the first original equation is:
4x + 3y = 13
replace x with 2.5 and y with 1 to gtet:
4(2.5) + 3(1) = 13 which becomes:
10 + 3 = 13 which is true.
the second original equation is:
2x - y = 4
replace x with 2.5 and y with 1 to get:
2(2.5) - 1) = 4 which becomes:
5 - 1 = 4 which is also true.
the solution applies to both equations and is confirmed as good.
the point where these lines intersect is (x,y) = (2.5,1).
that intersection point is the solution that is common to both equations.

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the mean of this data is equal to 4,541
the standard deviation is equal to 1008.
the point estimate of this data is the mean.
at 95% confidence level, the range of possible values you can expect in a sample of the same size 95% of the time would be equal to +/- 1.96 standard deviations about the mean which would be calculated as 4541 - 1008 * 1.96 to 4541 + 1008 * 1.96.
that would make the interval estimate from 2565.3 to 6516.7
the point estimate is 4541.
the standard deviation is 1008.

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x + y = z
x number of liters of 10% solution
y = number of liters of 1% solution.
z = number of liters of 2% solution.

.10x + .01y = .02z

the 2 equations you need to deal with are:
x + y = z (number of liters of each solution)
.10x + .01y = .02z (number of liters of acid in each solution).

you know that y = 25 because that's given.

your formulas becomes:
x + 25 = z
.10x + .01(25) = .02z

from the first equation, solve for z to get z = x + 25
replace z with x + 25 in the second solution to get:
.10x + .01(25) = .02(x + 25)
solve for x in this equation.
simplify the equaton to get:
.10x + .25 = .02x + .5
subtract .02x from both sides of this equation and subtract .25 from both sides of this equation to get:
.10x - .02x = .5 - .25
simplify to get:
.08x = .25
divide both sides of this equation by .08 to get:
x = .25 / .08 = 3.125

you need to add 3.125 liters of a 10% solution to 25 liters of a 1% solution to get a 2% solution.

since y = 25 and x = 3.125, then z must be equal to 28.125

x + y = z becomes 3.125 + 25 = 28.125
.10x + .01y = .02z becomes:
.10(3.125) + .01(25) = .02(28.125)
simplify this to get:
.3125 + .25 = .5625 which becomes:
.5625 = .5625

numbers check out so the solution is good.

you get .5625 liters of acid in the combined solution which is 2% of 28.125 liters.

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mean = 17.5
standard deviation is 4.4
you need to find the z-scores for 15 minutes, 18 minutes, 20 minutes and 25 minutes.
the z-score is equal to (x-m)/s
x is the raw score.
m is the mean.
s is the standard deviation.
exampl:
for 15 minutes, the z-score is (15 - 17.5) / 4.4
the z-scores are:
15 minutes = -.5
18 minutes = .18182
20 minutes = .63636
25 minutes = 1.77273
all rounded to the nearest 5 decimal places.
you then look up in the normal distribution tables to find the probability of getting a z-score greater than the indicated z-score.
from the tables, you get:
p(z > -.5) = .69146
p(z > .42786) = .42786
p(z > .26227) = .26227
p(z > 25) = .03814
translated to raw scores, these probabilities are the same as:
p(x > 15) = .69146
p(x > 18) = .42786
p(x > 20) = .26227
p(x > 25) = .03814
now you need to find:
p(x > 20 | x > 15)
p(x > 25 | x > 18)
the general formula is p(a | b) = p(a intersect b) / p(b)
translated to the first problem, this formula becomes:
p(x > 20 | x > 15) = p(x > 20 intersect x > 15) / p(x > 15).
p(x > 20 intersect x > 15) means that you have to wait greater than 15 minutes and you have to wait greater than 20 minutes at the same time. That happens when you are waiting greater than 20 minutes and not before.
Therefore, p(x > 20 intersect x > 15) must be p(x > 20).
that's my take on the intersect part of the equation.
the equation therefore becomes:
p(x > 20 | x > 15) = p(x > 20 intersect . > 15) / p(x > 15) = .26277 / .69146 which becomes equal to .37930
similarly, for the second problem, the formula becomes:
p(x > 25 | x > 18) = p(x > 25 intersect x > 18) / p(x > 18).
since p(x > 25 intersect x > 18) can only occur when x > 25, then this is equal to p(x > 25) which is equal to .03814.
the formula therefore becomes:
p(x > 25 | x > 18) = .03814 / .42786 = .08914
the formulas are given.
the main problem is in the interpretation of what constitutes a intersect b.
the logic looks sound so i'd go with this answer unless you have an answer from somebody else that you like better.
if it was me, i'd go with this one.
the alternate way to look at it is this.
suppose you went to the restaurant 100,000 times.
you would have had to wait greater than 15 minutes for 69,146 of those times.
you would have had to wait greater than 20 minutes for 26,227 of those times.
given that you had to wait greater than 15 minutes means your universe from which you select from is now 69,146 rather than 100,000.
the probability of waiting greater than 20 minutes given that you had to wait greater than 15 minutes becomes 26,227 / 69,146 which is equal to .37930
this means that, of the 69,146 times you had to wait greater than 15 minutes, you had to wait greater than 20 minutes 37.930 % of those times.

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http://tinypic.com/view.php?pic=14n0ifc&s=6
Angle B congruent to Angle C (alternate interior angles of parallel lines AB and CD)
Angle D congruent to Angle C (alternate interior angles of parallel lines BC and DE)
Angle B congruent to Angle D by the Transitive Property of Equality (If a = b and b = c, then a = c).

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log (x^2 + 5x + 16) = 1
since log(a) = b if and only if 10^b = a, your equations can be changed to be:
10^1 = x^2 + 5x + 16 which becomes:
x^2 + 5x + 16 = 10
subtract 10 from both sides of the equation to get:
x^2 + 5x + 6 = 0
the roots of that equation are at:
x = -2
x = -3
to confirm, substitute these values into the original equations to see if the original equation holds true.
the original equation is:
log (x^2 + 5x + 16) = 1
when you substitute -2 for x and when you substitute -3 for x, the equation holds true confirming the solution is good.
the solution to the equation is:
x = -2 or x = -3

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the lad that has won 4 rounds has won 20 cents.
the other lad has won 25 cents more so he has won 45 cents.
since 5 cents is equal to 1 round, the total number of rounds must be 4 + 9 = 13.

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solve the equations for y so you can graph them.
x + 2y > -2 is solved as follows:
subtract x from both sides of the equation to get:
2y > -2 + x
divide both sides of the equation by 2 to get:
y > (-2 + x) / 2
3x + y <= 3 is solved as follows:
subtract 3x from both sides of the equation to get:
y <= 3 - 3x
the 2 equations that you have modified so they can be graphed are:
y > (-2 + x) / 2
y <= 3 - 3x
the equations that you will graph are:
y = (-2 + x) / 2
y = 3 - 3x
those graphs will look like this:

the line that crosses the y-axis at y = 3 is the graph of the equation y = 3 - 3x.
the line that crosses the y-axis at y = -1 is the graph of the equation y = (-2 + x) / 2
now that you graphed the lines, you need to go back to the inequality equations and see what area you want to shade.
the inequality equations are:
y > (-2 + x) / 2
y <= 3 - 3x
you want to be above the line of the equation y = (-2 + x) / 2
you want to be below or on the line of the equation y = (3 - 3x)
you will need to show the line y = (-2 + x) / 2 as a dashed line because you will not be on the line, you will only be above it.
you will need to show the line y = 3 - 3x as a solid line because you will be on the line as well as below it.
the shaded area is shown in the picture of the graph shown below:

it is below and to the left of the line y = 3 - 3x.
it is above the line y = (-2 + x) / 2
it is only in the upper left section of the graph because that is the only section of the graph where both requirements are met.

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if it travels 25 miles in 50 seconds and there are 60 seconds in a minute, then it is traveling 25 / 50 * 60 = 30 miles per minute.
since there are 60 minutes in an hour,then it is traveling 30 * 60 = 1800 miles per hour.
that's pretty fast.