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Assuming that they are going to buy 1 film, then the amount of money they have left for books is $705 minus $30 = $675.

$675 / $3 = 225.

They will be able to buy 225 books with the money they have left over after they buy the film.

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An ellipse has a major axis and a minor axis.

In your equation, the major axis is horizontal and the minor axis is vertical.

The foci are at (0,-4) and (0,4).

The center of the ellipse is at the point (0,0).

c is the distance between each focus and the center of the ellipse.

This means that c = +/- 4.

This means that c^2 = 16

The major axis of the ellipse is equal to 2a.

This means that a is the distance from the center of the ellipse to the intersection of the major axis of the ellipse with the ellipse.

The minor axis of the ellipse is equal to 2b.

This means that b is the distance from the center of the ellipse to the intersection of the minor axis of the ellipse with the ellipse.

the major axis of this ellipse is vertical.

the minor axis of this ellipse is horizontal.

A diagram of the major axis of this ellipse and the minor axis of this ellipse and the location of the foci and the intersection of the major and minor axis of this ellipse with the ellipse is shown below.

everything is drawn except the ellipse itself which I can't do.

                                P(major)
                                x
                                F2
                            b   x
                            |   x
                            |   x
                 (minor)P x x x C x x x P(minor)
                          x     x
                  a -----   x   x   ----- c
                              x x
                                F1
                                x
                                P(major)

In the diagram above:

P(minor) is the intersection of the minor axis with the ellipse.
P(major) is the intersection of the major axis with the ellipse.
F1 is the focal point at (0,-4).
F2 is the focal point at (0,4).
C is the center of the ellipse at (0,0).

c is the straight line distance between C and F1.

a is the straight line distance between F1 and P(minor) on the left of the diagram (line not shown).

b is the straight line distance between C and P(minor) on the left of the diagram.

This forms a right triangle with the points of the triangle being F1, C, and P(minor) on the left of the diagram.

By the Pythagorean formula, a^2 = b^2 + c^2.

This formula leads to c^2 = a^2 - b^2.

Since c^2 = a^2 - b^2, and since c = 4, this means that:

16 = a^2 - b^2

The problem states that the sum of the focal radii = 10.

The sum of the focal radii is a constant and is equal to the distance between F1 and a point on the ellipse plus the distance between F2 and the same point on the ellipse.

By definition, the sum of these distances is equal to 2a which is the diameter of the major axis.

Since 10 is equal to 2a, this means that a = 5 which is the length from the center of the ellipse to the perimeter of the ellipse along the major axis.

In the diagram, that would be from C to P(major) on the top of the diagram.

In the diagram, it is also from C to P(major) on the bottom of the diagram.

Since we know that a = 5, and we know that c = 4, we can derive b using the formula:

c^2 = a^2 - b^2

This becomes

4^2 = 5^2 - b^2 which becomes:

16 = 25 - b^2.

We solve for b to get b = sqrt(9) = 3.

b is the length from the center of the ellipse to the ellipse along the minor axis.

In the diagram, that would be from C to P(minor) on the left of the diagram.

In the diagram, that would also be from C to P(minor) on the right side of the diagram.

We now have:

a = 5
b = 3
c = 4

From this, we should be able to derive the formula for the ellipse.

The standard formula for an ellipse is:

(x-h)^2 / b^2 + (y-k)^2 / a^2 = 1

Since the two foci are at (0,-4) and at (0,4), this means that the vertex of the ellipse (otherwise known as the center of the ellipse) is at (0,0).

This means that (h,k) = (0,0), because (h,k) represent the center of the ellipse (otherwise known as the vertex of the ellipse).

Since h is zero, then (x-h)^2 becomes x^2.

since k is zero, then (y-k)^2 becomes y^2.

Our formula becomes:

x^2 / b^2 + y^2 / a^2 = 1

Since a = 5 and b = 3, this formula becomes:

x^2 / 3^2 + y^2 / 5^2 = 1 which becomes:

x^2 / 9 + y^2 / 25 = 1

To graph this equation, we have to solve for y as follows:

Equation is:

x^2 / 9 + y^2 / 25 = 1

Subtract x^2 / 9 from both sides of the equation to get:

y^2 / 25 = -x^2/9 + 1

Multiply both sides of the equation by 25 to get:

y^2 = (-25/9)x^2 + 25

Take the square root of both sides of the equation to get:

y = +/- sqrt((-25/9)x^2 + 25)

This makes y = sqrt((-25/9)x^2 + 25) and y = -sqrt((-25/9)x^2 + 25)

Graph of this equation looks like this:



In this graph, the foci are at (x,y) = (0,-4) and at (x,y) = (0,4).
The major axis is vertical at y = 0.
The minor axis is horizontal at x = 0.
The vertex is at (x,y) = (0,0).
The distance from the foci to the vertex is +/- 4 along the y-axis.
The sum of the focal radii will always be 10.

The answer to your question is that the equation for this ellipse is:

x^2/9 + y^2/25 = 1

The general form of that equation is:

x^2/b^2 + y^2/a^2 = 1

A picture of this ellipse is shown below:





In this picture:
2a is the length of the major axis.
The designation for the distance from the surface of the ellipse to the center of the ellipse along the major axis is called "a".
2b is the length of the minor axis.
The designation for the distance from the surface of the ellipse to the center of the ellipse along the minor axis is called "b".
The designation for the distance from each focal point to the center of the ellipse is called "c".
F1 and F2 are the focal points of the ellipse.
The center of this ellipse is (0,0).
F1 is at (0,4) and F2 is at (0,-4).
*****
a also happens to be the hypotenuse of the right triangle formed by b and c as shown in the diagram. This is why the formula a^2 = b^2 + c^2 is valid.
Since a^2 = c^2 + b^2 is a valid equation (Pythagorean Formula for Right Triangle), the other formula that can be derived from this is:
c^2 = b^2 - a^2.
*****
What is not shown in this picture is the focal radii.
That will be shown in a separate picture below:





In this picture:

P1 is a point on the surface of the ellipse.
P2 is a point on the surface of the ellipse.
d11 is the distance from P1 to F1.
d12 is the distance from P1 to F2.
d21 is the distance from P2 to F1.
d22 is the distance from P2 to F2.
The focal radii shown on this ellipse are d11, d12, d21, d22.
From P1, the Sum of the focal radii are d11 and d12.
From P2, the Sum of the focal radii are d21 and d22.
The Sum of the focal radii is a contant.
This means that the sum will always be the same, regardless of which point on the surface of the ellipse you are at.
The Sum of d11 and d12 is equal to 10 for this ellipse.
The Sum of d21 and d22 is also equal to 10 for this ellipse.

A fairly decent reference regarding this problem can be found at the following website.

http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_ellipse.xml

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x^2 + y^2 - 10x + 4y + 13 = 0

This equations is not in standard form.

Standard form of the equation of a circle is:

(x-h)^2 + (y-k)^2 = r^2 where:

(h,k) are the coordinates of the center of the circle and r is the radius of the circle.

You need to convert your equation into standard form in order to solve this problem.

Move all the terms around until all the x terms are together and all the y terms are together, and the constant term is on the right side of the equation.

You get:

x^2 - 10x + y^2 + 4y = -13

Complete the squares for the x terms and y terms.

x^2 - 10x = (x-5)^2 - 25

y^2 + 4y = (y+2)^2 - 4

Your equation becomes:

(x-5)^2 - 25 + (y+2)^2 - 4 = -13

Add 25 and add 4 to both sides of this equation to get:

(x-5)^2 + (y+2)^2 = -13 + 29 which becomes:

(x-5)^2 + (y+2)^2 = 16

The center of your circle should be (x,y) = (5,-2)

The radius of your circle should be sqrt(16) = 4

Graph your circle to see if this is true.

To graph your circle, you need to solve for y.

Your equation to work with is:

(x-5)^2 + (y+2)^2 = 16

Subtract (x-5)^2 from both sides of this equation to get:

(y+2)^2 = -(x-5)^2 + 16

Take the square root of both sides of this equation to get:

y+2 = +/- sqrt(-(x-5)^2 + 16)

Subtract 2 from both sides of this equation to get

y = +/- sqrt(-(x-5)^2 + 16) - 2

You get 2 equations out of this.

They are:

y = + sqrt(-(x-5)^2 + 16) - 2
y = - sqrt(-(x-5)^2 + 16) - 2

Graph these 2 equations and you should have your circle.

The graph of these 2 equations is shown below:



I drew a horizontal line at y = -2.

Draw an imaginary vertical line at x = 5 and you'll see that the center of the circle is at (x,y) = (5,-2).

When y = -2, the edge of the circle is at x = 1, and at x = 9. Both of these are 4 units away from x = 5, proving that the radius of the circle is 4 units long.

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I don't really know.

But, .....

I did a search on the internet and found an article that indicated that the surface area of the pyramids was originally limestone.

Maybe they needed to determine how much limestone they needed?

Maybe somebody today wanted to know how much limestone they needed?

Here's the article.

http://arc.boardofstudies.nsw.edu.au/go/sc/maths/activities/the-pyramid-of-giza/

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I would say 16cm^2.

The radius of the circle is equal to 2cm.

The diameter of the circle is equal to 2 times the radius is equal to 4cm.

The diameter of the circle is equal to one side of the square since the circle is inscribed in the square.

The area of the square is equal to s^2 is equal to 4^2 is equal to 16cm^2.

Answer is selection d.

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In algebgra.com language, the answer is:

log(4,(64*256)) = log(4,64) + log(4,256)

After being put through the algebra.com formula generator, the answer looks like this:



In english, the answer is:

log(64*256) to the base of 4 is equal to log(64) to the base of 4 plus log(256) to the base of 4.

In your language, the answer is:

log[4](64*256) = log[4](64) + log[4](256).

This is because, in general:

log(b,a*c) = log(b,a) + log(b,c)

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graph of y = x^2 - 8x + 2 is shown below:



To plot the points on this graph, all you do is take values of x and then solve for y assuming that value of x replaces x in the equation.

example:

when x = 4, y = 4^2 - 8*4 + 2 = 16 - 32 + 2 = -16 + 2 = -14.

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expression is:

(2/9 + 4/9) * (1/3 - 9/10)

2/9 + 4/9 can be combined because their denominators are the same to get:

6/9

1/3 - 9/10 needs to find a common denominator for.

easiest one to find is 3 * 10 = 30.

multiply first fraction by 10/10 and multiply the second fraction by 3/3 to get:

10/30 - 27/30.

now that the denominator are the same, these can be combined to equal:

-17/30.

your expression of (2/9 + 4/9) * (1/3 - 9/10) is now equivalent to:

(6/9) * (-17/30) which can be multiplied together to get:

(6*-17)/(9*30) = -102/270

-102/270 = -.3777777778

Use your calculator to solve the original expression to see if it is the same as the result you get with the final expression.

They are the same so the simplification process was correct.

If you did not want to multiply the first expression by the second expression, then your answer would be:

first expression:

(2/9 + 4/9) is the same as 6/9 = 2/3

second expression:

(1/3 - 9/10) = (10/30 - 27/30) = -17/30

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2 * (1/4 + 1/5) + 2

1/4 is the same as 5/20
1/5 is the same as 4/20

expression becomes:

2 * (5/20 + 4/20) + 2

since the denominators are now the same, you can combine the fractions to get:

2 * (9/20) + 2

2 * 9/20 = 18/20 so your expression becomes:

18/20 + 2

18/20 can be simplified to 9/10 so your expression becomes:

2 and 9/10

2 is equivalent to 20/10.

your expression becomes 20/10 + 9/10.

Since the denominator is the same, these fractions can be combined to equal:

29/10 which equals 2.9

You can confirm this is accurate by using your calculator to determine the value of the original expression and see if it is the same as the value of the final expression.

I confirmed and they are the same so the answer is good.

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man insures building for $10,000 at annual rate of 1 and 1/2 percent per year.

That's 1.5% divided by 100% = a rate of .015 per year.

This means he would be paying .015*10000 = 150 per year.

He pays $1250 for a sprinkler.

Insurance company reduces the rate on his property to 1/4% per year.

That's .25% divided by 100% = a rate of .0025 per year.

This means he would be paying .0025*10000 = 25 per year.

The difference in what he would be paying per year would be 150 - 25 = 125 a year.

In other words, he would be saving 125 per year on the insurance rates he is paying because he bought the sprinkler.

1250 / 125 = 10.

He will make his money back in 10 years.

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The cost of health care 10 years ago was $2,000 per person per year.
The cost of health care today is $4,000 per person per year.
Based on a straight line projection, what do you estimate the cost of health care to be in 20 years?

You let x = the year.
You let y = the cost of health care per person.

x1 = -10 represents 10 years ago
y1 = $2000 represents the cost of health care per person 10 years ago.

you get (x1,y1) = (-10,2000)

x2 = 0 represents today.
y2 = $4000 represents the cost of health care per person today.

you get (x2,y2) = (0,4000)

You need to make an equation that relates health care to the year.

Your equation will be a linear equation of the form y = mx + b

m is the slope and b is the y-intercept (value of y when x = 0).

The slope is given by the formula m = y2-y1 / x2-x1

y2-y1 = 4000 - 2000 = 2000

x2-x1 = 0 - (-10) = 10

your slope is 2000/10 = 200 per year.

your y-intercept is the value of y when x = 0

your equation so far is y = 200*x + b

substitute one of the points (x1,y1) or (x2,y2) for x and y in the equation.

use (x2,y2) = 0, y = 4000

your equation becomes:

4000 = 2000*0 + b

solve for b to get b = 4000

your equation becomes y = 200*x + 4000

graph of this equation is shown below:



I placed horizontal lines at y = 2000, y = 4000, y = 8000

y = 2000 should correspond to the cost of health care 10 years ago. This is when x = -10.

y = 4000 should correspond to the cost of health care today. This is when x = 0.

y = 8000 should correspond to the cost of health care 20 years from now. This is when x = 20.

The points where these horizontal lines intersect with the graph of the equation y = 200*x + 4000 should correspond to x = -10, x = 0, and x = 20. Trace a vertical line down from those intersection points and you should intersect with the x-axis at about those values.

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2*log(7,3) + 3*log(7,2) = log(7,x)

We can rewrite this as:

log(7,x) = 2*log(7,3) + 3*log(7,2)

The rules of logarithms that apply here will be:

log(a^n) = n*log(a)
log(a*b) = log(a) + log(b)

First thing we notice is that 2*log(7,3) would be the same as log(7,3^2) which would be the same as log(7,9).

Second thing we notice is that 3*log(7,2) would be the same as log(7,2^3) which would be the same as log(7,8).

Our equation becomes:

log(7,x) = log(7,9) + log(7,8)

Third thing we notice is that log(7,9) + log(7,8) would be the same as log(7,9*8) which would be the same as log(7,72)

Our equation becomes:

log(7,x) = log(7,72)

This can only be true if x = 72.

Our original equation is:

log(7,x) = 2*log(7,3) + 3*log(7,2)

We replace x with 72 and we get:

log(7,72) = 2*log(7,3) + 3*log(7,2)

We can convert this equation to the base of 10 so we can solve using our calculator.

The conversion formula in general is:

log(b,x) = log(c,x)/log(c,b)

This reads as log of x to the base b = log of x to the base c divided by the log of b to the base c.

The conversion formula in our case is:

log(7,x) = log(10,x) / log(10,7)

Our equation becomes:

log(10,72/log(10,7) = 2*log(10,3)/log(10,7) + 3*log(10,2)/log(10,7)

If we multiply both sides of this equation by log(10,7), then we get:

log(10,72) = 2*log(10,3) + 3*log(10,2) which becomes:

1.857332496 = 2*.477121255 + 3*.301029996 which becomes:

1.857332496 = 1.857332496 which is true confirming the equations are equivalent.

Your answer is that x = 72.

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If the total budget for the 3 months was $252,000, and the monthly budget remained the same for the 3 months, then the monthly budget had to be $252,000 / 3 = $84,000 per month.

If they spent 90% of their monthly budget the first month, then they spent .9 * $84,000 = $75,600 the first month.

$84,000 - $75,600 = $8,400.

They spent $8,400 below their budget for the first month.

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let x = t.

your equation becomes V(x) = -16x^2 + vt + h

h = 50 which is the height of the rocket at time point 0 (value of x is 0).

your equation becomes v(x) = -16x^2 + vx + 50

Since initial velocity is 200 ft/sec, then v = 200 and your equation becomes:

v(x) = -16x^2 + 200x + 50

maximum height should be when x = -b/2a

a = -16
b = 200
c = 50

maximum height should be when x = -200/-32 = 6.25

when x = 6.25, v(x) = 675 feet.

That's what the maximum height should be.

Set v(x) = 300 and solve for x.

300 = -16x^2 + 200x + 50

subtract 50 from both sides of this equation to get 0 = -16x^2 + 200x - 250.

solve for the roots of this equation by quadratic formula.

a = -16
b = 200
c = -250

(-b +/- sqrt(b^2 - 4ac))/(2a) =:
(-200 +/- sqrt(200^2 - 4*-16*-200))/(-32) =:
(-200 +/- sqrt(40000 - 12800))/(-32) =:
(-200 +/- sqrt(27200)))=32 =:
(-200 +/- 164.924225)/(-32)

x = 1.096117968 or x = 11.40388203

The rocket should be at or above 300 feet from x = 1.096117968 to x = 11.40388203

A graph of this equation is shown below:



A horizontal line was placed at y = 300 and 675 to show you the intersection points with the graph of the equation of the rocket.

At y = 675, the graph should be at the maximum point.

At 300, the graph should be at x = 1.09 and x = 11.4 roughly.

Trace a vertical line down from those intersection points to see the x-value.

x-value represents time in seconds.
y-value represents height in feet.

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circumference = 2*pi*r

circumference = 1 revolution

1 revolution = 2*pi*r

r = 5/pi

1 revolution = 2*pi*5/pi = 2*5 = 10

2 revolutions = 20

Answer is selection c.

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-14

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100 - .2*100 = 100 - 20 = 80

80 + .2*80 = 80 + 16 = 96

Answer is selection a.

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let the volume of the first glass be x
let the volume of the second glass be 2x
the volume of the container is 3x.

The amount of alcohol in the first glass is (1/2)*x
The amount of alcohol in the second glass is (1/3)*2x = (2/3)*x
The total amount of alcohol in the container is (1/2)*x + (2/3)*x = (3/6)*x + (4/6)*x = (7/6)*x
The total volume of the container is 3*x.
The fraction of alcohol in the container is therefore (7/6)*x / (3*x) which is equal to (7/18).

As an example:

Assume the first glass is 6 ounces.
It is half full of alcohol so this contains 3 ounces of alcohol.

The second glass is twice the volume of the first glass so the volume of the second glass is 12 ounces.
It is 1/3 full of alcohol so this contains 4 ounces of alcohol.

The total alcohol in both glasses is 3 ounces + 4 ounces = 7 ounces.

The total volume of the mixture in the container is 6 ounces + 12 ounces = 18 ounces.

The fraction of alcohol in the container is 7 ounces / 18 ounces = 7/18

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Find sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6)

It looks like all of these revolve around a 30/60/90 triangle.

sin(30) = 1/2
cos(30) = sqrt(3)/2
tan(30) = 1/sqrt(3)
cot(30) = sqrt(3)
sin(60) = sqrt(3)/2
cos(60) = 1/2
tan(60) = sqrt(3)
cot(60) = 1/sqrt(3)

Since it's easier to look at these in degrees, then you can convert radians to degrees by using the following formula:

degrees = radians * pi * 180 / pi.

Your original expression is:

sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6)

-5pi/3 * 180/pi = -5*180/3 = -300 degrees
2pi * 180/pi = 360 degrees
-13pi/3 * 180/pi = -13*180/3 = -780 degrees
-29pi/6 * 180/pi = -29*180/6 = -870 degrees

If you add 360 to -300 degrees, you get an equivalent angle of 60 degrees that is in the first quadrant.
If you subtract 360 from 360 degrees, you get an equivalent angle of 0 degrees that is in the first quadrant.
If you add 2 * 360 degrees to -780 degrees, you get an equivalent angle off -60 degrees which has a reference angle of 60 degrees that is in the fourth quadrant.
If you add 3 * 360 to -870 degrees, you get an angle of 210 degrees which has a reference angle of 30 degrees that is in the third quadrant.

your original expression of:

sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6) becomes:

sin(60)cos(0)tan(-60)cot(210)

sin(60) = sqrt(3)/2)
cos(0) = 1
tan(-60) = -tan(60) = -sqrt(3)
cot(210) = cot(30) = sqrt(3)

Now all you have to do is multiply them together.

You can also verify with your calculator.

Using the calculator in radian mode, I got the following:

sin(-5pi/3) = .866025404 which is the same as sqrt(3)/2.
cos(2pi) = 1.
tan(-13pi/3) = -1.732050808 which is the same as -sqrt(3).
cot(-29pi/6) = 1.732050808 which is the same as sqrt(3).

This agrees with the answers I got in degree mode so it's looking good.

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This would happen is m = n and b = c.

This means that the lines are identical.

If the lines are identical then they intersect at all points.

If the equations are in same standard form, then this would happen is the equations are multiples of each other.

An example:

5x + 7y = 35
10x + 14y = 70

To convert these equations into slope intercept form, we solve for y.

The first equation becomes:

5x + 7y = 35 is the original equation.
Subtract 5x from both sides of this equation to get:
7y = -5x + 35
Divide both sides of this equation by 7 to get:
y = (-5/7)x + (35/7)

the second equation becomes:
10x + 14y = 70 is the original equation.
Subtract 10x from both sides of the equation to get:
14y = -10x + 70
Divide both sides of the equation by 14 to get:
y = (-10/14)x + (70/14)
Simplify to get:
y = (-5/7)x + (35/7)

The equations that were multiples of each other in standard form become the same as each other in slope-intercept form.

If you graph both these equations, you will get the same line.

If you get the same line, then there are an infinite number of solutions that will satisfy both equations.

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Your equation is:

h = 96t - 16t^2

You want to solve this equation for when h = 0.

Your equation becomes:

-16t^2 + 96t = 0

Divide both sides of this equation by 16 to get:

-t^2 + 6t = 0

Factor out the t to get:

-t * (t-6) = 0

This equation will be true if -t = 0 or (t-6) = 0 or both are 0.

Solve for t and you get:

t = 0 or t = 6.

Graph your original equation of -16t^2 + 96t = 0 as shown below:



Note that in order to graph this equation, you needed to substitute x for t so your equation becomes y = -16x^2 + 96x.

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Your answer is selection E (none of the above).

He needs to do 10 laps at an average speed of 60 miles per hour.

Since Rate * Time = Distance, this means that 60 * T = 10 because:

Average rate needs to be 60 miles per hour.
2 laps at 5 miles per lap = total of 10 miles.

Solve for T and you get T = 10/60 = 5/30.

On the first lap, he traveled at 30 miles per hour to cover 5 miles.

Since R*T = D, this means that:

30*T = 5 which means that T = 5/30.

Since he used up all of his available time on the first lap, he will never be able to average 60 miles per hour on the second lap because he has 0 time left in which to do it in.

If you want to use the Rate * Time = Distance Formulas a different way, it would come out like this:

30*T1 = 5 where T1 is the time it takes for him to do the first lap.

60*(T1 + T2) = 10 here T1 is the time it takes for him to do the first lap and T2 is the time it takes for him to do the second lap.

Since 30*T1 = 5, then T1 = 5/30 so we can susbtitute for T1 in the second equation to get:

60*(5/30 + T2) = 10

We divide both sides of this equation by 60 to get:

5/30 + T2 = 10/60

We subtract 5/30 from both sides of this equation to get:

T2 = 10/60 - 5/30

We multiply 5/30 by 2/2 to get 10/60

Out equation becomes:

T2 = 10/60 - 10/60 = 0

In order for him to average 60 miles per hour overall, T2 has to be 0 which is impossible.

If x = rate he requires on the second lap, then formula for second lap becomes:

x*T2 = 5

If we divide both sides of this equation by T2, then we get x = 5/T2.

Since T2 = 0, then this equation is invalid and there is no value of x that will satisfy it.

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QUESTION 1:

8 + (-2) - (-6) = 8 - 2 + 6 = 6 + 6 = 12.

8 + (-2) is the same as subtracting 2 from 8 to get 6.

6 - (-6) is the same as adding 6 to 6 to get 12.

positive plus negative is the same as positive minus positive with the sign of the result coming from the larger operand.
positive minus negative is the same as positive + positive with the sign of the result always being positive.

QUESTION 2:

3*(-6) + (-7)*(-10) = -18 + 70 = 52.

3*(-6) will always be negative because a plus times a minus is always minus.

(-7)*(-10) will always be positive because a minus times a minus is always positive.

-18 plus 70 is the same as 70 minus 18 = 52.

positive times negative will always be negative.
negative times negative will always be positive.
negative plus positive is the same as positive plus negative is the same as positive minus positive with the sign of the result coming from the larger operand.

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h = husband's age
d = daughter's age.

Your husband is 4 times your daughter's age now>

This is represented by the formula:

h = 4d

Your husband will be 2 times as old as your daughter in 20 years.

This is represented by the formula:

(h+20) = 2(d+20)

Since h = 4d from the first equation, then substitute 4d for h in the second equation to get:

(4d + 20) = 2(d+20)

you now have one equation in one unknown that you can solve.

simplify to get:

4d + 20 = 2d + 40

subtract 2d from both sides of the equation and subtract 20 from both sides of this equation to get:

2d = 20

divide both sides of the equation by 2 to get:

d = 10

Your daughter is 10 years old now.

Since h = 4d, then your husband is 40 years old now.

In 20 years your daughter will be 30

In 20 years your husband will be 60.

In 20 years your husband will be twice your daughter's age.

Your answer is that the father is 40 years old now.

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Your answer is that the number of hole punchers is 5.

You can't find it directly because you have 2 equations in 3 unknowns which means that one of the unknowns can only be found in relation to one of the other unknowns.

To find your solution, I did the following:

The two equations you have to solve simultaneously are:

x + y + z = 100
.5x + 3y + 10z = 100

Multiply the second equation by 2 to get:

x + y + z = 100
x + 6y + 20z = 200

Subtract first equation from second equation to get:

5y + 19z = 100

Solve y in terms of z to get:

y = (100 - 19z)/5

This is the same as:

y = 100/5 - 19z/5

y will not be an integer unless z = 5 or a multiple of 5 which means that the smallest z can be is 5.

Sounds like she could buy either 5 or 10 hole punchers and y would be an integer.

If z were 10, however, then she could not have bought any x or y because 10 * 10 = 100 which means zero money for x and y.

z would have to be equal to 5.

Here's how the equations work out.

x + y + 5 = 100
.5x + 3y + 50 = 100

Subtract 5 from both sides of the first equation and subtract 50 from both sides of the second equation to get:

x + y = 95
.5x + 3y = 50

Multiply the second equation by 2 to get

x + y = 95
x + 6y = 100

Subtract the first equation from the second equation to get:

5y = 5

Divide both sides of the equation by 5 to get:

y = 1

Use the first equation to solve for x to get:

x = 94

Your values for x,y,z are 94,1,5.

Plug them into your original equations to get:

x + y + z = 100 becomes 94 + 1 + 5 = 100 which is true.
.5x + 3y + 10z = 100 becomes .5*94 + 3*1 + 10*5 = 47 + 3 + 50 = 100 which is true.

Both original equations are true when you use the values for x,y,z of 94,1,5, so those values are good.

Your answer is:

She bought 5 hole punchers.

that would be selection E.

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49 < n < 101
19 < m < 51

m and n are both whole numbers.

equation is (n+m)/n

greatest value for this equation will be when the numerator is maximum and the denominator is minimum.

minimum value for n is 50.
maximum value for n is 100
maximum value for m is 50

maximum value of (m+n)/n would be 150/50 = 3

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Your equation is:

15x^2 - 26x + 8

Your factors are:

(5x-2) * (3x-4)

If you multiply these factors out, you will get:

5x*(3x-4) - 2*(3x-4) which becomes:

5x*3x - 5x*4 - -2*3x + 2*4 which becomes:

15x^2 - 20x - 6x + 8 which becomes:

15x^2 - 26x + 8 which is your original equation.

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x = number of sets of dishes that cost $20 per set.
y = number of sets of dishes that cost $45 per set.
Total number of sets of dishes is equal to 250.
Total amount of money you have to spend is equal to $6800.

You have 2 equations that have to be solved simultaneously.

The first equation is the number of sets of dishes and is equal to:

x + y = 250

The second equation is the total money you have to spend on each set of dishes that will add up exactly to $6800 and is equal to:

20*x + 45*y = 6800

You can solve these 2 equations by either substitution or addition method.

We'll use substitution.

Take the first equation and solve for x or y.

we'll solve for x to get x = 250 - y

Substitute this value for x in the second equation to get:


20*x + 45*y = 6800 becomes 20*(250-y) + 45*y = 6800.

Simplify to get:

5000 - 20*y + 45*y = 6800

Simplify further to get:

5000 + 25*y = 6800

Subtract 5000 from both sides of this equation to get:

25*y = 1800

Divide both sides of this equation by 25 to get:

y = 72

Since x + y = 250, then x = 178

You need to buy 178 sets of dishes at $20 per set and 72 sets of dishes at $45 per set in order for you to buy a total of 250 sets of dishes and spend a total of $6800 exactly.

178 * 20 + 72 * 45 = 6800 confirming these values are good.

Your answer is:

You need to purchase 178 sets of dishes at $20 per set, and 72 sets of dishes at $45 per set.

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A person pays $98.10 as annual tax on his house and lot. If the rate of it is 1-1/2% and the value of the property is 75% of the actual cost, what is the cost of the property? The answer is supposed to be $8,720.00

x = actual cost of the property.
.75 * x = value of the property.

Person pays $98.10 annual tax on his property.
This tax is based on the value of the property.
Tax rate is .015 * value of the property.

Your equation is:

98.10 = .015 * .75 * x

Simplify to get:

98.10 = .01125 * x

Divide both sides of this equation by .01125 to get:

x = $8720.00

This agrees with the value that you're supposed to get so it's good.

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C = Cost of Phonograph = $200.
M = Marked Up Price of Phonograph = (1+x) * C.
D = Discounted Selling Price of Phonograph = 1.125 * C.
D = Discounted Selling Price of Phonograph = .9 * M.

Note that M - .1 * M = .9 * M.

You have 2 equations that need to be solved simultaneously.

They are:

D = 1.125 * C and D = .9 * M.

Since M = (1+x) * C, Then the equations become:

D = 1.125 * C and D = .9 * (1+x) * C.

Since both expressions equal to D, then both expressions are equal to each other and you get:

1.125 * C = .9 * (1+x) * C

Divide C from both sides of this equation to get

1.125 = .9 * (1+x)

Divide both sides of this equation by .9 and you get:

1.25 = 1 + x

Subtract 1 from both sides of this equation and you get

.25 = x

That's your answer.

The Price needs to be marked up 25% in order to sell it at a discount of 10% and still make a profit of 12.5%


In numbers, this is what happens:

The cost is $200.
the marked price is 1.25 * 200 = $250.
The discounted price is $250 - .1 * $250 = $250 - $25 = $225.
The profit is $225 - $200 = $25.
The profit margin is $25/$200 = .125 = 12.5%.