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a = bus ready for service
b = working radio

p(a) = probability of a = .83
p(a+b) = probability of a and b occurring simultaneously = p(a) * p(b) = .69

p(b|a) = probability of b given a.

p(b|a) = p(a+b) / p(a).

p(b|a) = .69 / .83 = .831325301

That's your answer.

If you look at it in terms of percentages, this is what is happening.

Assume there are 100 buses in total.

83 are in service.

69 out of the 83 have a working radio.

The percentage of buses ready for service that have a working radio is therefore 69/83 = .831325301 * 100% = 83.1325301%.

Since 83.1325301% of the buses ready for service have a working radio, the probability of getting a bus that has a working radio, given that the bus you have is ready for service, is .831325301.

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If the number of degrees per minute is in radians, then the formula should be:

velocity = angular velocity * radius

If that's true, then:

angular velocity = 400 radians per minute.
radius = 160 centimeters.

velocity = 400 * 160 = 64000 centimeters per minute.

Since there are 60 minutes in an hour, then this is equivalent to 3,840,000 centimeters per hour.

Since there are 100,000 centimeters in a kilometer, then this is equivalent to 3.84 kilometers per hour.

This formula assumes that the angular velocity is expressed in radians per unit of time (minutes in this case).

If you did not know the formula, then you would have had to rationalize the problem as follows:

In general, the linear velocity is equal to the number of rotations of the wheel per minute times the circumference of the wheel.

The circumference of the wheel is equal to 2*pi*r = 2*pi*160 = 1005.309649 centimeters.

Since 1 complete rotation of the wheel is equal to 2*pi radians (360 degrees), then the number of rotations of the wheel per minute is equal to the number of radians the wheel is spinning per minute divided by 2*pi.

400/2*pi = 63.66197724 rotations of the wheel per minute.

63.66197724 * 1005.309649 = 64000 centimeters of linear travel per minute.

Since there are 60 minutes in an hour, then this is equivalent to 3,840,000 centimeters per hour.

Since there are 100,000 centimeters in a kilometer, then this is equivalent to 3.84 kilometers per hour.

You could also have converted everything to degrees and gotten the same answer as follows:

Since number of degrees is equal to number of radians * 180 / pi, then 400 radians per minute of rotation is equivalent to 22918.31181 degrees per minute of rotation.

Number of rotations of the wheel per minute is equal to number of degrees of rotation per minute / 360 = 63.66197724 rotations of the wheel per minute.

That gets you the same answer.

You just had to know that the linear velocity of the wheel is equal to the number of rotations of the wheel per minute times the circumference of the wheel.

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If you make 1992 = 0, the 1996 should equal to 4 because 1996 - 1992 = 4 and 4 - 0 = 4.

You have 2 points of data.

They are:

(0,69.1)
(4,72.9)

If you let x = number of years and y = age expectancy, then you have an equation that looks like:

y = m*x + b

The slope of this equation is m which would be the change in life expectancy divided by the change in years.

Using your 2 points of data, this would become:

(72.9 - 69.1) / (4 - 0) = 3.8 / 4 = .95

This means that the life expectancy increases by .95 years every year.

4 * .95 = 3.8.

You start with 69.1 in year 0 and 4 years later you have 69.1 + 3.8 = 72.9

Now that you have the slope, you can fill that in the equation to get:

y = .95*x + b

b is the y-intercept.

You already know that b will be 69.1, but you can also solve for it to confirm.

You can pick either of the 2 points to replace x and y with in the equation to solve for b.

pick the point (0,69.1).

y = .95*x + b becomes 69.1 = .95*0 + b which becomes b = 69.1.

Your equation becomes y = .95*x + 69.1

Now you can set y equal to e(x) which makes your equation equal to e(x) = .95*x + 69.1

If you want to change x to t, then your equation becomes e(t) = .95*t + 69.1

Either way, that's your equation.

The key to solving this problem is to determine from the data that is given, the change in life expectancy per year.

That was done when we solved for the slope of the equation of the straight line above.

The slope was the change in life expectancy divided by the change in years which resulted in the change in life expectancy per year.

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Unfortunately, this problem depends on how much you made in the first quarter.

Since the revenue in the rest of the year is 5 times the revenue in the first quarter, if you did not make at least 30,000 in the first quarter, you would not make enough revenue during the rest of the year to cover the minimum expenditure of 150,000.

The graph will show that.

You would plot the revenue in the first quarter along the x-axis, and the revenue in the rest of the year on the y-axis.

The equation would be y = 5*x

I made the scale of my graph in thousands (10 = 10,000, 150 = 150,000, etc.).

You would then plot another equation at y = 150 (thousand).

To meet the constraints of your problem, x would have to be greater than or equal to 30 (thousand), and y would have to be greater than or equal to 150 (thousand).

Your graph would look like this:



There are 2 lines on the graph that you can see.

The third line is x = 30 which is a vertical line that can't be drawn on this graph but can be drawn on a manually generated graph.

Your shaded area would be in the region bounded by the lines

x = 30
y = 150,000
y = 5*x

The values you are looking for would be in the region where the value of x is >= 30, and the value of y >= 150,000 and the value of y <= 5*x.

You can see from the graph that when x < 30, the amount of revenue would not be sufficient to cover the minimum budget expenditure of 150,000.

Since x represents the amount of money made in the first quarter, then that translates to the amount of money made in the first quarter needs to be equal to or above 30,000 in order for the minimum budgetary requirements to be met from revenue, assuming that the amount of money to be made in the rest of the year is 5 times the amount of money made in the first quarter.

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Choose a number at random from 1 to 7.
Toss a coin.
Choose a letter at random from the word SCHOOL.
None of the above.

I believe the third choice would not have an equally likely probability for each of the letters to be picked.

For the first choice, the numbers are 1,2,3,4,5,6,7 with equal probability to pick any one of them because there is only one of each number.

There are 7 letters and each has a probability of 1/7 to be picked.

For the second choice, the possibilities are heads or tail with equal probability to pick any one of them because there is only one of each possibility.

There are 2 possible outcomes and each has a probability of 1/2 to be picked.

With the word SCHOOL, the probability for the letter O to come up is two times the probability for any of the other letters which have only one of each.

Total letters are 6.

Probability of S = 1/6
Probability of C = 1/6
Probability of H = 1/6
Probability of O = 2/6 *****
Probability of L = 1/6

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Use the direct variation formula of y = k*x

Solve for k first using the known values of x and y.

Let x = 2 and y = 8.4

Formula becomes:

8.4 = k*2

Divide both sides of this equation by 2 to get:

k = 4.2

k winds up being the slope of your straight line equation.

The formula for a straight line equation is y = m*x + b

m is the slope and b is the y-intercept.

The slope is 4.2.

The y-intercept is found by substituting known values for x and y in the equation and solving for b.

the known values are (x,y) = (2,8.4)

Equation of y = 4.2*x + b becomes 8.4 = 4.2*2 + b which becomes 8.4 = 8.4 + b.

Solve for b to get b = 0.

Your straight line equation is y = 4.2*x

Graph this equation to get:



Horizontal line at y = 8.4 intersects the graph of the equation of the line at x = 2.

Horizontal line at y = 14.7 intersects the graph of the equation of the lin at x = 3.5.

This confirms the equation is good.

When x = 2, y = 8.4

When x = 3.5, y = 14.7

The rate of change is the slope of the line which is 4.2.

This means that for every change of 1 in the value of x, y changes by 4.2.

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Basic concepts you will use:







(exponent is )



Start with each term separately.

Your expression is:



First term becomes:

= =

Second term becomes:

= = =

Third term becomes:

= =

Putting all 3 terms together, your expression becoms:

- +

Since , then your expression becomes:



You can confirm the final expression is equivalent to the original expression by taking any values of x, y, and z and substituting in both the original expression and the final expression.

If you come up with the same answer, you did good.l

I confirmed with x = 3, y = 4, and z = 5.

I used rather than because the calculator can solve logs to the base of 10 more easily.

That shouldn't matter since the expressions should work with any base as long as all bases in the expression are the same.

Just to make sure, I solved with as well.

Both confirmed that the answer with the original expression was the same as the answer with the final expression.

When you solve an expression to the base of anything other than 10 using the calculator, you simply take the take the log of the expression to the base of 10 and then divide that answer by the log of the other base to the base of 10.

An example will show you how that's done.

You know that = 32 because you can use your calculator to confirm that.

This means the = 5 because that's the way logs work.

By definition, = 5 if and only if = 32.

You can't solve that log directly using your calculator, but you can solve the log using the base conversion formula.

You get:

= = 5

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The general equation for a horizontally oriented hyperbola would be:



The center of your hyperbola is at C = (-2,0).

The vertices of this hyperbola are probably at V1 = (-2,-3) and V2 = (-2,3)

This, I believe, makes this a vertically oriented hyperbola whose general equation would be:



The y = +/- .5x refers to the equation of the asymptotes for the hyperbola.

In a horizontally oriented hyperbola, that equation would be:

y = +/-

In a vertically oriented hyperbola, that equation would be:

y = +/-

The distance between the center of the hyperbola and each vertex of the hyperbola is equal to "a".

Since your center is (-2,0) and one of your vertices is at (2,3), then "a" must be equal to 3.

Since the general equation of the asymptotes of a vertically oriented hyperbola is:

y = +/- and the equation of the asymptotes of your equation is y = +/- , then the equation of the asymptotes for your hyperbola becomes:

y = +/- .5x = +/-

This makes:

+/- .5x = +/-

The slope of the asymptote on the left side of this equation is +/- .5

The slope of the asymptote on the right side of this equation is +/- .

Since a = +/- 3, this makes the slope of the asymptote on the right side of this equation equal to

Because these slopes are just different versions of the slope of the same asymptotes, we can solve for b as follows:

+/- .5 = +/-

Multiply both sides of this equation by b to get:

+/- .5b = +/- 3

Multiply both sides of this equation by 2 to get:

+/- b = +/- 6

We now have:

a = 3
b = 6

Since the general form of the equation of our vertically oriented hyperbola is:

, we can replace a^2 with 9 and b^2 with 36 to get:



Since the center of our hyperbola is at (-2,0), and the center of the hyperbola is represented by (h,k), this means that:

h = -2
k = 0

We can replace h and k in the equation of our hyperbola to get:



That's the equation of our hyperbola.

To graph this equation, we have to solve for y.

Solving for y gets us:

y = +/-

The graph of our equation looks like this:



A more distant view looks like this:



The equation for the asymptotes of this hyperbola are y = +.- .5

We add the equations of those lines to the distant graph to get:



The asymptotes are a little off because we did not account for the center of the hyperbola.

Apparently, the equation y = +/- .5x was the equation for the slope of the asymptotes only.

Since the asymptotes have to go through the center of the hyperbola, we need to modify this equation to account for the fact that the lines are going through the point (-2,0).

The point slope form of the equation of a straight line is (y-y1) = m*(x-x1)

The slope of our line is +/- .5

The point slope form of the equation of the asymptotes of our hyperbola becomes:

(y-y1) = +/- .5*(x-x1)

Since the lines goes through the point (-2,0), then (x1,y1) = (-2,0) and the equation of our asymptotes becomes:

y = +/- .5*(x+2)

We graph these equations of our asymptotes on top of the equation of our hyperbola to get:



Now everything lines up right and the graph clearly shows the asymptotes of the equation of our hyperbola.

You did not need to graph the asymptotes. That's a bonus.

Another bonus is the eccentricity of the hyperbola.

It is given by the equation

c is the measure of the distance from the center of the hyperbola to one of the foci of the hyperbola.

The value of c is given by the equation

In this hyperbola, that would make

The eccentricity of this hyperbola is therefore = 2.236067978

The higher the eccentricity, the straighter is the hyperbola. A very very high eccentricity would make the hyperbola look more like two parallel lines.

This one is somewhere in the medium range.

"a" and "c" are defined as:

a is the distance from the center of the hyperbola to each vertex.
c is the distance from the center of the hyperbola to each focus.

A pretty decent reference for you to look at is shown below:

http://www.purplemath.com/modules/hyperbola.htm

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g(x) = -(1/5)x+1

You solve for x and then you invert x and y as follows:

Let y = g(x).

Equation becomes:

y = -(1/5)*x + 1

Solve for x as follows:

Subtract 1 from both sides of the equation to get:

y - 1 = -(1/5)*x

Multiply both sides of the equation by 5 to get:

5 * (y-1) = -x

Multiply both sides of the equation by -1 to get:

-5 * (y-1) = x

Simplify to get:

x = -5y + 5

Invert x and y in this equation to get:

y = -5x + 5

That should be your inverse equation.

Replace y with h(x) to get:

h(x) = -5x + 5

You could have used any letter other than h(x). a(x), b(x), c(x) would have done just as well. I just chose h(x) arbitrarily because that letter wasn't used yet and it was the next letter in line in the alphabet.

g(x) is your equation.

h(x) is your inverse equation.

If h(x) is the inverse equation, you should be able to see that h(x) and g(x) are reflections about the line y = x.

Graph both of these equations plus equation of y = x to see if that's true.

Graph is shown below:



The line in the middle going from bottom left to top right is the line of y = x.

The line with the very steep slope going from top left to bottom right is the line of y = -5x + 5.

The line with the shallow slope going from top left to bottom right is the line of y = -(1/5)x + 1.

It's a little hard top see but they look like they're reflections about the line y = x (mirror images).

The other way to tell is because the inverse function undoes what the function does.

What this results in is that h(g(x)) = x

Here's how that works.

g(x) = -(1/5)*x + 1

Let x = any real number.

We'll choose 160.

Replace x in g(x) with 160 to get:

g(160) = -(1/5)*(160) + 1 which equals -31

h(x) = -5x + 5

Replace x in h(x) with (-31) that we just calculated for g(x) to get:

h(g(x)) = h(-31) = -5*-31 + 5 which becomes 155 + 5 which equals 160.

g(160) = -31
h(-31) = 160

h(x) undid what g(x) did which makes h(x) the inverse function of g(x).

You could also have shown that h(g(x)) = x by just solving the equations in x without substituting numbers for x.

Here's how you would have done that:

g(x) = -(1/5)*x + 1

h(x)= -5x + 5

h(g(x)) = -5 * (-(1/5)*x + 1) + 5

You replaced x in h(x) with g(x) and you replaced x in (-5x + 5) with (-(1/5)*x + 1)

Simplify by distributing the multiplication to get:

h(g(x)) = (-5 * (-1/5) * x) - (5 * 1) + 5

Since -5 * (-1/5) * x) = x, and - (5 * 1) = - 5, your equation becomes:

h(g(x)) = x - 5 + 5

Combine like terms to get:

h(g(x)) = x

That shows h(x) is the inverse function of g(x).

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When you shade 2 parts out of 12 parts in the circle, the shaded part is 2/12 of the circle.

When you shade 2 parts out of 3 parts of the same circle ( or a congruent circle), the shaded part is 2/3 of the circle.

You can compare the fractions if the denominators are the same.

If you multiply the numerator and the denominator of a fraction by the same number, then the value of the fraction remains the same.

1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 6/12

In your problem:

2/3 = 4/6 = 6/9 = 8/12

2/3 is the same fraction as 8/12.

Matt split his circle into 12 equal parts and shaded 2 of them.

Matt's fraction is 2/12.

Mark split his circle into 3 equal parts and shaded 2 of them.

Mark's fraction is 2/3 which is the same as 8/12.

Mark's fraction is 4 times the size of Matt's fraction.



To compare fractions, it helps to get the fractions to have the same denominator.

In order to change the denominator of a fraction, you would multiply the numerator and the denominator by the same factor (the number 4 in this example).



In order to change it back, you would divide the numerator and the denominator by the same factor that you multiplied them by (the number 4 in this example).



If you wish to visualize this, then the following diagram might help.





Just pretend this is a circle rather than a dodecagon (12 sided regular polygon).

Matt shaded 2/12 of his circle. That's represented by the 2 dots in the left hand sided figure.

Mark shaded 2/3 of his circle. That's represented by the 8 dots in the right hand sided figure.

2/3 is the same as 8/12.

Each dot is 1/12.

4 dots make up 1/3 because (4*1)/(4*3) = 4/12.

8 dots make up 2/3 because (2*4*1)/(4*3) = 8/12.

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Here's a graph of a circle.



Here's a graph of the same circle.



In the first graph it looks like a circle.

In the second graph it looks like an ellipse.

All I did was change the scale of the vertical axis (the y-axis).

In the first graph the x and y axis spanned from -10 to 10.

In the second graph, the x axis spanned from -10 to 10 and the y axis spanned from -20 to 20.

If you are not aware of the scaling of each axis, you can definitely be misled by what's on the graph.

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Pythagorean Theorem is c^2 = a^2 + b^2

c is the hypotenuse and a and b are the legs of the right triangle.

The hypotenuse is always bigger than either of the 2 legs.

You are given that one side of the triangle is equal to sqrt(17).

You are also given that the second side of the triangle is equal to sqrt(5).

sqrt(5) cannot be the hypotenuse of the right triangle.

sqrt(17) could.

sqrt(x) could also, since you do not know what size that is.

Assuming that x is the hypotenuse of the triangle, then the Pythagorean formula would be:

x^2 = 17 + 5

c^2 = x^2
a^2 = 17
b^2 = 5

Simplify to get:

x^2 = 23

That would make x = sqrt(23).

Assuming that 17 is the hypotenuse of the triangle, then the Pythagorean formula would be:

17 = 5 + x^2

c^2 = 17
a^2 = 5
b^2 = x^2

Subtract 5 from both sides of that equation to get:

x^2 = 17 - 5

Combine like terms to get:

x^2 = 12

That would make x = sqrt(12).

Either answer would be correct if it was not specified which side was which.

In a right triangle, the Pythagorean Formula holds:

c^2 = a^2 + b^2

With our first triangle, this becomes:

5 + 17 = 23

With our second triangle, this becomes:

5 + 12 = 17

One of these will be your correct answer once you determine which side was supposed to be 5 and which side was supposed to be 17.

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A perfect square is an integer that is the result of the squaring of another integer.

25 would be a perfect square because it is the result of 5*5

169 would be a perfect square because it is the result of 13*13

the factors of the perfect square can be positive or negative.

Example

5*5 = 25

(-5)*(-5) = 25

Both 5 and -5 are square roots of the perfect square of 25.

In your problem:

square root of (2*40) = square root of 80 = 8.94427191 (not an integer)

square root of (5*40) = square root of 200 = 14.14213562 (not an integer)

square root of (20*40) = square root of 800 = 28.28427125 (not an integer)

square root of (40*40) = square root of 1600 = 40 (this is the perfect square)

Answer is selection D.

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Graph the quadratic function
f(x) = -x^2 + 1
Describe the correct graph and why...Which way does the parabola go up or down, where does the graph cross the x-axis and y-axis....

The graph points up and opens down.

Standard form of quadratic equation is ax^2 + bx + c = 0

Set f(x) = to 0 and you have this equation in standard form.

You get:

-x^2 + 1 = 0

In this equation:

a = -1
b = 0
c = 1

Maximum point is at x = -b/2a which becomes 0

When x = 0, y = 1, so the maximum point is (x,y) = (0,1).

To find the points where this graph crosses the x-axis, you have to solve the equation -x^2 + 1 = 0

With this equation, you subtract 1 from both sides of the equation to get:

-x^2 = -1

Multiply both sides of this equation by -1 to get:

x^2 = 1

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

x = +/- 1

Those should be the x-axis crossing points.

You could also have factored the equation of -x^2 + 1 = 0 to get:

(-x+1) * (x+1) = 0

When either of these factors = 0, the equation is grue, so you set each of the factors equal to 0 and solve.

You get:

x = -1 and x = 1.

You graph this equation by plotting some values of x and getting corresponding values of y.

You start with x = -1, x = 0, x = 1

That should be enough to draw a rough graph, but you might want to fill in some additional points to fit the curve better.

Your graph should look like this:

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First problem:

(y^2 - 8) * (4y^2 - 4y + 7) equals:

y^2 * (4y^2 - 4y + 7) - 8 * (4y^2 - 4y + 7) which equals:

4y^4 - 4y^3 + 7y^2 - 32y^2 + 32y - 56

Combine like terms to get:

4y^4 - 4y^3 - 25y^2 + 32y - 56

Second problem:

16b^6 - 2p^6

You can factor out a 2 to get:

2 * (8b^6 - p^6)

Third problem:

(x^5 - 6) + (x^5 + 6) equals:

x^5 - 6 + x^5 + 6 after removing parentheses.

Combine like terms to get:

2x^5.

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Rate * Time = Distance

D = 54 miles.

Total distance is the distance traveled by the first boat plus the distance traveled by the second boat.

The time traveled by both boats will be the same.

We call it T.

the distance traveled by both boats is 54 miles.

The distance traveled by the first boat is equal to x (we assigned that).

the distance traveled by the second boat is equal to y (we assigned that also).

x + y = 54.

The rate of the first boat is 10 miles an hour.

the rate of the second boat is 8 miles an hour.

We have 2 equations that need to be solved simultaneously.

The first equation is:

10 * T = x

the second equation is :

8 * T = y

We also know that x + y = 54

Since x = 10 * T and y = 8 * T, we can susbtitute in the equation of x + y = 54 to get:

10 * T + 8 * T = 54

This means that 18 * T = 54 which means that T = 3.

Substituting for T in our equation gets:

10 * 3 + 8 * 3 = 54 which becomes 30 + 24 = 54 which becomes 54 = 54 which is true.

The boats will be 54 miles apart in 3 hours.

The first boat will have traveled 30 miles in 3 hours at 10 miles per hour.

The second boat will have traveled 24 miles in 3 hours at 8 miles per hour.

Since they are going in opposite directions, the total distance between them will be 30 + 24 = 54 miles in 3 hours.

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We start with the circular table and determine the radius of the circle formed by the table.

The area is given as 1256 square inches.

the formula for the area of a circle is .

This makes square inches.

this makes inches.

Since the diameter of a circle equals 2 times the radius, then the diameter of the table = 39.98985957 inches.

Each leaf has a width of 12 inches and a length of the diameter of the circle which would be a length of 39.98985957 inches

The area of each of the leafs will be 12 inches * 39.98985957 inches = 479.8783148 square inches.

Since the new square inches is 2696 and the old square inches is 1256, then the number of additional square inches of area is equal to 2696 - 1256 = 1440 square inches.

1440 / 479.8783148 = 3.000760725 additional leafs required.

Since you can only add leafs in integral amounts, the answer would be 3 leafs.

3 leafs would give you an area of 1439.634944 additional square inches on top of 1256 square inches which would give you a total area of 2695.634944 square inches.

This is close to the area you were looking for but not right on.

If you round to the nearest integer, however, then the answer would be correct.

3 leafs added to the existing area of the table would become 2696 square inches.

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Box contains 40 cookies.
24 of them are round.
20 of them are chocolate.
12 are neither round nor chocolate.

If 24 are round, then 16 must be not round.
If 20 are chocolate, then 20 must be not chocolate.

The count is therefore:

24 round and 16 not round
20 chocolate and 20 not chocolate

Since we have 12 not round and not chocolate, we'll move them to their own category to get:

24 round and 4 not round
20 chocolate and 8 not chocolate
12 not round and not chocolate

The 4 remaining that are not round must be chocolate. Otherwise they would have been counted in the not round and not chocolate.

We'll move them to their own category to get a revised count of:

24 round
16 chocolate and 8 not chocolate
12 not round and not chocolate
4 not round and chocolate

The 8 that are not chocolate must be round. Otherwise they would have been counted in the not round and not chocolate also.

We'll move them to their own category to get a revised count of:

16 round
16 chocolate
12 not round and not chocolate
4 not round and chocolate
8 round and not chocolate = 8

The 16 that are round and the 16 that are chocolate must be the same cookies.

We'll move them to their own category to get a revised count of:

12 not round and not chocolate = 12
4 not round and chocolate = 4
8 round and not chocolate = 8
16 round and chocolate = 16

The total number of cookies is equal to 12 + 4 + 8 + 16 = 40 as it should be.

Your answer is that the number of cookies that are round and chocolate is 16.

That would be selection C.

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Rate * Time = Units

Rate of water pouring into the container is 4 liters per minute.

Rate of water being pumped out of the container is 5 liters per minute.

The net rate of water leaving the container is equivalent to 1 liter per minute.

When there are 50 liters of water in the container, the equation becomes:

1 * Time = 50

Time = 50 minutes.

It would take 50 minutes to empty the container assuming that water is pouring in at 4 liters per minute and water is being pumped out at 5 liters per minute.

Look at it from the perspective of the actual rates and you'll see how this works.

There are 50 liters of water in the tank.

Water is pouring in at 4 liters per minute for the next 50 minutes.

50 * 4 = 200 liters on top of the 50 liters that are already in the tank making a total of 250 liters.

In the same 50 minutes, the pump is taking water out of the container at 5 liters per minute.

50 * 5 = 250 liters of water being pumped out of the container.

This leaves the container empty after 50 minutes since the same amount of water that was pouring in, plus the amount of water that was already in the containter, has been pumped out.

The equation that we used was 1 * T = 50 which made T = 50.

T represents Time.

We could have used another equation as follows:

4*T + 50 = 5*T

When the values are equal, the tank is empty because 4*T + 50 is the number of liters of water coming in (we start at 50), and 5*T is the number of liters of water going out.

Solve for this equation to get:

T = 50.

Same answer.

In 50 minutes, the tank will be empty.

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Speed of the slower plane = x miles per hour.
Speed of the faster plane = (x+50) miles per hour.

Rate * Time = Distance.

Faster plane equation becomes:

(x+50) * 3 = D

Slower plane equation becomes:

x * 3.5 = D

Since they both equal D, then they both equal each other and we get:

(x+50) * 3 = x * 3.5

Simplify to get:

3*x + 150 = 3.5*x

Subtract 3*x from both sides of the equation to get:

.5*x = 150

Divide both sides of the equation by .5 to get:

x = 300

The slower plane is flying at 300 miles per hour.

The faster plane is traveling at 350 miles per hour.

the equation for the faster plane becomes:

350 * 3 = D = 1050 miles.

The equation for the slower plane becomes:

300 * 3.5 = D = 1050 miles.

the distance is the same in both equations as it should be.

The distance is 1050 miles.

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Equation of the circle is x^2 + y^2 = 169

The point (12,5) is on the circle.

Since the general equation for a circle is (x-h)^2 + (y-k)^2 = r^2, and (h,k) is the center of the circle, this means that:

The center of the circle is at (h,k) = (0,0)

The radius of the circle is sqrt(169) = 13

The line tangent to the circle at the point (12,5) is perpendicular to the radius of the circle at that point.

the equation for the line that is identical to the radius of the circle (the radius of the circle is a segment of this line bounded by the circumference of the circle) would be given by the equation:

y = m*x + b where m is the slope and b is the y-intercept.

The slope of this line is given by the equation:

(y2-y1) / (x2-x1).

y2 = 5 and y1 = 0
x2 = 12 and x1 = 0

slope of the line = m = (5-0)/(12-0) = 5/12

The equation of this line becomes y = (5/12)*x + b

The y-intercept of this line is found by replacing x and y in the equation by one of the points of the equation and solving for b.

We'll use (x2,y1) = (12,5)

We get:

y = (5/12)x + b becomes 5 = (5/12)*12 + b

Simplify to get:

5 = 5 + b

Subtract 5 from both sides of this equation to get:

b = 0

the y-intercept for this equation is 0 and the equation is:

y = (5/12)*x

This line is identical to the radius of the circle which means that the radius of the circle is part of this line.

The line itself extends indefinitely in both directions.

the radius is a line segment bounded by the circumference of the circle.

The equation of the line perpendicular to that radius will have a slope that is equal to the negative reciprocal of the slope of that line.

This means that the line that represents the tangent to the circle will have a slope of (-12/5)

Since that line passes through the point (12,5), then you can find the y-intercept of that line by replacing y and x in the equation for that line with 5 and 12 to get:

y = (-5/12)x + b becomes 5 = (-12/5)*12 + b

Simplify to get:

5 = -144/5 + b

Multiply both sides of this equation by 5 to get:

25 = -144 + 5b

Add 144 to both sides of this equation to get:

25 + 144 = 5b

Combine like terms to get:

169 = 5b

Divide both sides of this equation by 5 to get:

b = 169/5

the equation for the tangent to the circle passing through the point (12,5) would be:

y = (-12/5)*x + 169/5

169/5 is roughly 33.8 for graphing purposes.

To graph the equation of the circle itself, you have to solve for y.

The equation of the circle is x^2 + y^2 = 169

Solve for y to get y = +/- sqrt(169-x^2)

To graph the circle, you would graph 2 equations.

They would be:

y = sqrt(169-x^2) and y = -sqrt(169-x^2).

Since the equations of the lines are already in slope-intercept form which is the same for required for graphing them, then no change needs to be made to those equations.

You will therefore graph the following equations:

y = (5/12)*x
y = (-12/5)*x + (169/5)
y = sqrt(169-x^2)
y = -sqrt(169-x^2)

The graph of these equations is shown below:



You can see that the line identical to the radius of the circle passes through the points (0,0) and (12,5) and (-12,-5).

The radius is bounded by the points (12,5) and (-12,-5). The line itself extends indefinitely in both directions.

You can see that the line tangent to the circle at the point (12,5) is perpendicular to the radius of the circle at that point and passes through the points (12,5) and (0,33.8). The y-intercept of that line is the point (0,33.8).

A closer look at the tangent point is shown below:

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16r^3 - 9r = 0

Factor out the r on the left side of the equation to get

r * (16r^2 - 9) = 0

r = 0 will satisfy this equation because anything times 0 equals 0.

16r^2 - 9 = 0 will also satisfy this equation because anything times 0 equals 0.

Solve for 16r^2 - 9 = 0

Add 9 to both sides of this equation to get 16r^2 = 9

Divide both sides of this equation by 16 to get r^2 = (9/16)

Take the square root of both sides of this equation to get r = +/- sqrt(9/16)

Since sqrt(9/16) is the same as sqrt(9) / sqrt(16), and since sqrt(9) = 3, and since sqrt(16) = 4, you get r = +/- 3/4

You have 3 possible answers to this equation.

They are:

r = 0
r = 3/4
r = -3/4

Plug those values into the original equation to see if they hold up.

Your original equation is 16r^3 - 9r = 0

When r = 0, this equation becomes 0 - 0 = 0 which is true.

When r = 3/4, this equation becomes 16*27/64 - 9*3/4 = 432/64 - 27/4 = 6.75 - 6.75 = 0 which is true.

When r = -3/4, this equation becomes 16*-27/64 + 9*3/4 = -432/64 + 27/4 = -6.75 + 6.75 = 0 which is true.

All 3 answers are solutions to the original equation.

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You need 30 tons of rock to cover the area.

That is fixed.

30 * 60 = $1800.

You have $2500 to spend.

$2500 - 1800 = $700.

You have $700 to spend on trees.

$700 / $84 = 8.333333333 trees.

You can buy up to 8 trees.

the formula you would use is as follows:

Let T = number of trees.
Let R = number tons of rocks.
Let x = price per ton of rocks.
Let y = price per tree.

the general formula you would use is

T <=

When x = $60 and R = 30 and y = $84, this formula becomes:

T <=

Simplify this equation to get:

T <=

Simplify further to get:

T <=

Simplify further to get:

T <= 8.333333333

Since you can't buy a partial tree, you can buy up to 8 trees.

"<=" means less than or equal to

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Assume it was only 3 answers on the test and the student needed to get at least 2 out of 3 correct.

Probability of getting exactly 0 wrong would be .5^3 = .125 * 1 = .125
Probability of getting exactly 1 wrong would be .5^3 = .125 * 3 = .375
Probability of getting exactly 2 wrong would be .5^3 = .125 * 3 = .375
Probability of getting exactly 3 wrong would be .5^3 = .125 * 1 = .125

Total probability is equal to 1 as it should be.

Probability of getting 0 or 1 wrong would be .375 + .125 = .5

Since 0 or 1 wrong is the same as getting 2 or 3 right, then this is the probability that the student will get at least 2 right.

The individual probabilities are multiplied by the number of ways they can occur.

If we let 0 = wrong and 1 = correct, then:

You can get 0 wrong only 1 way (111)
You can get 1 wrong 3 ways (110) (101) (011)
You can get 2 wrong 3 ways (001) (010) (100)
You can get 3 wrong 1 way (000)

The same concept applies to the larger numbers.

-----

With 10 answers, this is what happens:

p(0) = probability of getting exactly 0 correct.
p(1) = probability of getting exactly 1 correct.
etc.

p(0) = .5^10 = .000976563 * 1 = .000976563
p(1) = .5^10 = .000976563 * 10 = .009765625
p(2) = .5^10 = .000976563 * 45 = .043945313
p(3) = .5^10 = .000976563 * 120 = .1171875
p(4) = .5^10 = .000976563 * 210 = .205078125
p(5) = .5^10 = .000976563 * 252 = .24609375
p(6) = .5^10 = .000976563 * 210 = .205078125
p(7) = .5^10 = .000976563 * 120 = .1171875
p(8) = .5^10 = .000976563 * 45 = .043945313
p(9) = .5^10 = .000976563 * 10 = .009765625
p(10) = .5^10 = .000976563 * 1 = .000976563

Total probability equals 1 as it should.

Probability of getting 0 or 1 or 2 wrong is equal to:

.000976563 + .009765625 + .043945313 = .0546875

Since the probability of getting 0 or 1 or 2 wrong is the same as the probability of getting 8 or 9 or 10 right, then the probability that the student will get at least 8 correct is equal to .0546875.

The number of ways each percentage can be achieved is given by the formula:



For example, with 10 answers, the number of ways of getting exactly 4 wrong is equal to:



With 10 answers, the number of ways of getting exactly 6 wrong is the same as getting exactly 4 wrong as shown below:

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Your system of equations would be as follows:

x = the number of pounds of the first kind of jelly beans.
y = the number of pounds of the second kind of jelly beans.

Total of 53 pounds of jelly beans was sold.

Equation 1 would be:

x + y = 53

The first kind of jelly beans was priced at $4.45 per pound.
The second kind of jelly beans was priced at $1.12 per pound.
The total amount of money made was equal to $125.96

Equation 2 would be:

4.45 * x + 1.12 * y = 125.96

The two equations you have to solve simultaneously are:

x + y = 53 (equation 1)
4.45 * x + 1.12 * y = 125.96 (equation 2)

You can solve by substitution or by elimination.

Either way will get you the same answer.

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Surface area of a cylinder is equal to (2 times the area of the base) plus (the area of the height times the circumference of the cylinder).

Unfortunately, the ratio of the surface area will vary depending on the height.

I'll do an example to show you what I mean.

The surface area of the base of the cylinder is equal to .

The surface area of the cylinder itself is equal to

The total surface area is given by the equation:

S =

Assume the radius is equal to 2.

You would get:

S2 = which would become:

S2 =

Assume the radius is equal to 3.

You would get:

S3 = which would become:

S3 =

When the ratio of the radii is 2:3, your formulas become:

S2 =
S3 =

If we assume h is equal to 5, then we would get:

S2 =
S3 =

which would become:

S2 =
S3 =

which would become:

S2 =
S3 =

If we assume that h = 20, then we would get:

S2 =
S3 =

which would become:

S2 =
S3 =

which would become:

S2 =
S3 =

28/48 = .583333333
88/138 = .637681159

28/48 is not the same ratio as 88/138 so the height makes a difference.

A reference for the formula of the surface area of a cylinder is shown below:

http://www.aaamath.com/geo79x10.htm

Your problem does not appear to be worded correctly.

It needs to give the height as well, or it needs to be asking for something else, like the volume of the cylinder.

In that case, the equation for the volume of the cylinder is equal to:

V = .

You would get:

V2 = and V3 = and the ratio between V2 and V3 would be equal to 4/9.

That would remain the same regardless of the height of the cylinder and regardless of the radius of the cylinder as long as the ratio between the radii of the cylinders was the same.

This is because the and the h both cancel out in this ratio.

= which becomes:

= which becomes:

= = after the and the and the cancel out.

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The student was required to get 45% of the maximum marks to pass.

The student failed by 15% of the maximum marks.

This means that the student got 45% of the maximum marks minus 15% of the maximum marks which equals 30% of the maximum marks.

If we let M equal the maximum marks that the student can get, then the equation would be:

Student got:

.45*M - .15*M = (.45 - .15)*M = .30*M.

Since the student got a score of 138 marks, then our equation becomes:

.30*M = 138

Divide both sides of this equation by .30 to get:

M = 138/.3

Solve for M to get M = 460.

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=

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The largest angle has to be smaller than 160 degrees.

If the largest angle is 160 degrees, then the angle that it is 8 times larger than is equal to 160/8 = 20 degrees.

The sum of these 2 angles is 180 degrees which forces the third angle to be 0 degrees which is not allowed since each angle of the triangle has to be greater than 0 degrees.

So the largest angle has to be less than 160 degrees.

You will never be able to find the largest angle because, no matter how close you can get to 160 degrees, you will always be able to get a little closer.

It's simpler to say that the largest angle has to be smaller than 160 degrees.

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p = pounds of peanuts
c = pounds of cashews
3p = cost for peanuts
4c = cost for cashews

Your equations are:

p + c = 100 (you need 100 pounds of mixed nuts)
3p + 4c = x (it will cost you x number of dollars)

You need to find a value for x which you will be able to spend.

x will have to be greater than or equal to $300 (all peanuts).
x will have to be less than or equal to $400 (all cashews).

Assume x is $375.00

This means that you can spend $375.00 on mixed nuts.

Your formulas become:

p + c = 100
3p + 4c = 375

You would then solve these 2 equations simultaneously to find out how many pounds of peanuts and cashews you can buy.

Multiply the first equation by 3 to get:

3p + 3c = 300
3p + 4c = 375

Subtract the first equation from the second equation to get:

c = 75

Since p + c = 100, this means that p = 25.

Total cost would be 25*3 + 75*4 = 75 + 300 = $375.00.

With $375.00, you can buy 25 pounds of peanuts and 75 pounds of cashews.

If you only had $325 to spend, then your equations would have been:

p + c = 100
3p + 4c = 325

Multiply first equation by 3 to get:

3p + 3c = 300
3p + 4c = 325

Subtract first equation from second equation to get:

c = 25

If c = 25, then p = 75, because p + c = 100

Total cost would be 75*3 + 25*4 = 225 + 100 = $325.00

If you had $325.00 to spend, you could buy 75 pounds of peanuts and 25 pounds of cashews.

So:

With $375 to spend, you could buy 25 pounds of peanuts and 75 pounds of cashews.
With $325 to spend, you could buy 75 pounds of peanuts and 25 pounds of cashews.

The amount of money you had to spend between $300 and $400 determines how many pounds of peanuts you can buy versus how many pounds of cashews you can buy in order to get 100 pounds of mixed nuts.

If you had less than $300 to spend, you would not be able to buy 100 pounds, no matter what mix you chose.

If you had more than $400 to spend, you would have money left over, no matter what mix you chose.

You only needed between $300 and $400 in order for the pounds to be equal to 100 pounds and no money to be left over.

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These are all solutions of the equation and possible solutions of the problem.

This does not mean that they are all actual solutions to the problem.

You would reject x <= 0 as not being a viable solution because the solution has to be a positive number.