How to use the inverse variation formula - SAT Math

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Question

The square of varies inversely with the cube of . If when , then what is the value of when ?

Answer

When two quantities vary inversely, their products are always equal to a constant, which we can call k. If the square of x and the cube of y vary inversely, this means that the product of the square of x and the cube of y will equal k. We can represent the square of x as x2 and the cube of y as y3. Now, we can write the equation for inverse variation.

x2y3 = k

We are told that when x = 8, y = 8. We can substitute these values into our equation for inverse variation and then solve for k.

82(83) = k

k = 82(83)

Because this will probably be a large number, it might help just to keep it in exponent form. Let's apply the property of exponents which says that abac = ab+c.

k = 82(83) = 82+3 = 85.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 85 in for k.

x2y3 = 85

(1)2y3 = 85

y3 = 85

In order to solve this, we will have to take a cube root. Thus, it will help to rewrite 8 as the cube of 2, or 23.

y3 = (23)5

We can now apply the property of exponents that states that (ab)c = abc.

y3 = 23•5 = 215

In order to get y by itself, we will have the raise each side of the equation to the 1/3 power.

(y3)(1/3) = (215)(1/3)

Once again, let's apply the property (ab)c = abc.

y(3 • 1/3) = 2(15 • 1/3)

y = 25 = 32

The answer is 32.

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Question

varies directly as and inversely as .

$z = y ^{2}$ and $B = \sqrt{A}$ .

Which of the following is true about ?

Answer

varies directly as and inversely as , so for some constant of variation ,

$\frac{xy}{A} =K$ .

We can square both sides to obtain:

$\left (\frac{xy}{A} \right ) ^{2}=K^{2}$

$\frac{x^{2}y^{2}}{A^{2}}= K^{2}$

$z = y ^{2}$ .

$B = \sqrt{A}$ , so $B^{4} =\left ( \sqrt{A} \right )^{4} = A^{2}$ .

By substitution,

$\frac{x^{2}z}{B^{4}}= K^{2}$ .

Using $K^{2}$ as the constant of variation, we see that varies inversely as the square of and directly as the fourth power of .

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Question

The radius of the base of a cylinder is ; the height of the same cylinder is ; the cylinder has volume 1,000.

$t = \sqrt{r}$

$v = \sqrt[4]{h}$

Which of the following is a true statement?

Assume all quantities are positive.

Answer

The volume of a cylinder can be calculated from its height and the radius of its base using the formula:

$\pi r^{2} h = V$

$t = \sqrt{r}$ , so $t ^{2}= r$ ;

$v = \sqrt[4]{h}$ , so $h = v^{4}$ .

The volume is 1,000, and by substitution, using the other equations:

$\pi (t^{2})^{2} v^{4} = 1,000$

$\pi t^{4} v^{4} = 1,000$

$\frac{\pi t^{4} v^{4}}{\pi} = \frac{1,000}{\pi}$

$t^{4} v^{4} = \frac{1,000}{\pi}$

$\sqrt[4]{t^{4} v^{4} }= \sqrt[4]{\frac{1,000}{\pi}}$

$tv= \sqrt[4]{\frac{1,000}{\pi}}$

If we take $K= \sqrt[4]{\frac{1,000}{\pi}}$ as the constant of variation, we get

,

meaning that varies inversely as .

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Question

varies directly as the square of and the cube root of , and inversely as the fourth root of . Which of the following is a true statement?

Answer

varies directly as the square of and the cube root of , and inversely as the fourth root of , so, for some constant of variation ,

$\frac{s \sqrt[4]{w}}{y^{2}\sqrt[3]{t}} = K$

We take the reciprocal of both sides, then extract the square root:

$\frac{y^{2}\sqrt[3]{t}}{s \sqrt[4]{w}} =\frac{1} { K}$

$\frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} =\frac{1} { K}$

$\left ( \frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} \right )^{\frac{1}{2}} =\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y t^{\frac{1}{6}}}{s ^{\frac{1}{2}} w ^{\frac{1}{8}}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y\sqrt[6]{t}}{\sqrt{s} \cdot \sqrt[8]{w}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

Taking $\left (\frac{1} { K} \right )^{\frac{1}{2}}$ as the constant of variation, we see that varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

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Question

If varies inversely as , and when , find when .

Answer

The formula for inverse variation is as follows: $y=\frac{k}{x}$

Use the x and y values from the first part of the sentence to find k.

$14=\frac{k}{3}$

Then use that k value and the given x value to find y.

$y=\frac{42}{28}$

$y=1\frac{1}{2}$

Compare your answer with the correct one above

Question

The square of varies inversely with the cube of . If when , then what is the value of when ?

Answer

When two quantities vary inversely, their products are always equal to a constant, which we can call k. If the square of x and the cube of y vary inversely, this means that the product of the square of x and the cube of y will equal k. We can represent the square of x as x2 and the cube of y as y3. Now, we can write the equation for inverse variation.

x2y3 = k

We are told that when x = 8, y = 8. We can substitute these values into our equation for inverse variation and then solve for k.

82(83) = k

k = 82(83)

Because this will probably be a large number, it might help just to keep it in exponent form. Let's apply the property of exponents which says that abac = ab+c.

k = 82(83) = 82+3 = 85.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 85 in for k.

x2y3 = 85

(1)2y3 = 85

y3 = 85

In order to solve this, we will have to take a cube root. Thus, it will help to rewrite 8 as the cube of 2, or 23.

y3 = (23)5

We can now apply the property of exponents that states that (ab)c = abc.

y3 = 23•5 = 215

In order to get y by itself, we will have the raise each side of the equation to the 1/3 power.

(y3)(1/3) = (215)(1/3)

Once again, let's apply the property (ab)c = abc.

y(3 • 1/3) = 2(15 • 1/3)

y = 25 = 32

The answer is 32.

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Question

varies directly as and inversely as .

$z = y ^{2}$ and $B = \sqrt{A}$ .

Which of the following is true about ?

Answer

varies directly as and inversely as , so for some constant of variation ,

$\frac{xy}{A} =K$ .

We can square both sides to obtain:

$\left (\frac{xy}{A} \right ) ^{2}=K^{2}$

$\frac{x^{2}y^{2}}{A^{2}}= K^{2}$

$z = y ^{2}$ .

$B = \sqrt{A}$ , so $B^{4} =\left ( \sqrt{A} \right )^{4} = A^{2}$ .

By substitution,

$\frac{x^{2}z}{B^{4}}= K^{2}$ .

Using $K^{2}$ as the constant of variation, we see that varies inversely as the square of and directly as the fourth power of .

Compare your answer with the correct one above

Question

The radius of the base of a cylinder is ; the height of the same cylinder is ; the cylinder has volume 1,000.

$t = \sqrt{r}$

$v = \sqrt[4]{h}$

Which of the following is a true statement?

Assume all quantities are positive.

Answer

The volume of a cylinder can be calculated from its height and the radius of its base using the formula:

$\pi r^{2} h = V$

$t = \sqrt{r}$ , so $t ^{2}= r$ ;

$v = \sqrt[4]{h}$ , so $h = v^{4}$ .

The volume is 1,000, and by substitution, using the other equations:

$\pi (t^{2})^{2} v^{4} = 1,000$

$\pi t^{4} v^{4} = 1,000$

$\frac{\pi t^{4} v^{4}}{\pi} = \frac{1,000}{\pi}$

$t^{4} v^{4} = \frac{1,000}{\pi}$

$\sqrt[4]{t^{4} v^{4} }= \sqrt[4]{\frac{1,000}{\pi}}$

$tv= \sqrt[4]{\frac{1,000}{\pi}}$

If we take $K= \sqrt[4]{\frac{1,000}{\pi}}$ as the constant of variation, we get

,

meaning that varies inversely as .

Compare your answer with the correct one above

Question

varies directly as the square of and the cube root of , and inversely as the fourth root of . Which of the following is a true statement?

Answer

varies directly as the square of and the cube root of , and inversely as the fourth root of , so, for some constant of variation ,

$\frac{s \sqrt[4]{w}}{y^{2}\sqrt[3]{t}} = K$

We take the reciprocal of both sides, then extract the square root:

$\frac{y^{2}\sqrt[3]{t}}{s \sqrt[4]{w}} =\frac{1} { K}$

$\frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} =\frac{1} { K}$

$\left ( \frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} \right )^{\frac{1}{2}} =\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y t^{\frac{1}{6}}}{s ^{\frac{1}{2}} w ^{\frac{1}{8}}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y\sqrt[6]{t}}{\sqrt{s} \cdot \sqrt[8]{w}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

Taking $\left (\frac{1} { K} \right )^{\frac{1}{2}}$ as the constant of variation, we see that varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

Compare your answer with the correct one above

Question

If varies inversely as , and when , find when .

Answer

The formula for inverse variation is as follows: $y=\frac{k}{x}$

Use the x and y values from the first part of the sentence to find k.

$14=\frac{k}{3}$

Then use that k value and the given x value to find y.

$y=\frac{42}{28}$

$y=1\frac{1}{2}$

Compare your answer with the correct one above

Question

The square of varies inversely with the cube of . If when , then what is the value of when ?

Answer

When two quantities vary inversely, their products are always equal to a constant, which we can call k. If the square of x and the cube of y vary inversely, this means that the product of the square of x and the cube of y will equal k. We can represent the square of x as x2 and the cube of y as y3. Now, we can write the equation for inverse variation.

x2y3 = k

We are told that when x = 8, y = 8. We can substitute these values into our equation for inverse variation and then solve for k.

82(83) = k

k = 82(83)

Because this will probably be a large number, it might help just to keep it in exponent form. Let's apply the property of exponents which says that abac = ab+c.

k = 82(83) = 82+3 = 85.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 85 in for k.

x2y3 = 85

(1)2y3 = 85

y3 = 85

In order to solve this, we will have to take a cube root. Thus, it will help to rewrite 8 as the cube of 2, or 23.

y3 = (23)5

We can now apply the property of exponents that states that (ab)c = abc.

y3 = 23•5 = 215

In order to get y by itself, we will have the raise each side of the equation to the 1/3 power.

(y3)(1/3) = (215)(1/3)

Once again, let's apply the property (ab)c = abc.

y(3 • 1/3) = 2(15 • 1/3)

y = 25 = 32

The answer is 32.

Compare your answer with the correct one above

Question

varies directly as and inversely as .

$z = y ^{2}$ and $B = \sqrt{A}$ .

Which of the following is true about ?

Answer

varies directly as and inversely as , so for some constant of variation ,

$\frac{xy}{A} =K$ .

We can square both sides to obtain:

$\left (\frac{xy}{A} \right ) ^{2}=K^{2}$

$\frac{x^{2}y^{2}}{A^{2}}= K^{2}$

$z = y ^{2}$ .

$B = \sqrt{A}$ , so $B^{4} =\left ( \sqrt{A} \right )^{4} = A^{2}$ .

By substitution,

$\frac{x^{2}z}{B^{4}}= K^{2}$ .

Using $K^{2}$ as the constant of variation, we see that varies inversely as the square of and directly as the fourth power of .

Compare your answer with the correct one above

Question

The radius of the base of a cylinder is ; the height of the same cylinder is ; the cylinder has volume 1,000.

$t = \sqrt{r}$

$v = \sqrt[4]{h}$

Which of the following is a true statement?

Assume all quantities are positive.

Answer

The volume of a cylinder can be calculated from its height and the radius of its base using the formula:

$\pi r^{2} h = V$

$t = \sqrt{r}$ , so $t ^{2}= r$ ;

$v = \sqrt[4]{h}$ , so $h = v^{4}$ .

The volume is 1,000, and by substitution, using the other equations:

$\pi (t^{2})^{2} v^{4} = 1,000$

$\pi t^{4} v^{4} = 1,000$

$\frac{\pi t^{4} v^{4}}{\pi} = \frac{1,000}{\pi}$

$t^{4} v^{4} = \frac{1,000}{\pi}$

$\sqrt[4]{t^{4} v^{4} }= \sqrt[4]{\frac{1,000}{\pi}}$

$tv= \sqrt[4]{\frac{1,000}{\pi}}$

If we take $K= \sqrt[4]{\frac{1,000}{\pi}}$ as the constant of variation, we get

,

meaning that varies inversely as .

Compare your answer with the correct one above

Question

varies directly as the square of and the cube root of , and inversely as the fourth root of . Which of the following is a true statement?

Answer

varies directly as the square of and the cube root of , and inversely as the fourth root of , so, for some constant of variation ,

$\frac{s \sqrt[4]{w}}{y^{2}\sqrt[3]{t}} = K$

We take the reciprocal of both sides, then extract the square root:

$\frac{y^{2}\sqrt[3]{t}}{s \sqrt[4]{w}} =\frac{1} { K}$

$\frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} =\frac{1} { K}$

$\left ( \frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} \right )^{\frac{1}{2}} =\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y t^{\frac{1}{6}}}{s ^{\frac{1}{2}} w ^{\frac{1}{8}}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y\sqrt[6]{t}}{\sqrt{s} \cdot \sqrt[8]{w}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

Taking $\left (\frac{1} { K} \right )^{\frac{1}{2}}$ as the constant of variation, we see that varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

Compare your answer with the correct one above

Question

If varies inversely as , and when , find when .

Answer

The formula for inverse variation is as follows: $y=\frac{k}{x}$

Use the x and y values from the first part of the sentence to find k.

$14=\frac{k}{3}$

Then use that k value and the given x value to find y.

$y=\frac{42}{28}$

$y=1\frac{1}{2}$

Compare your answer with the correct one above

Question

The square of varies inversely with the cube of . If when , then what is the value of when ?

Answer

When two quantities vary inversely, their products are always equal to a constant, which we can call k. If the square of x and the cube of y vary inversely, this means that the product of the square of x and the cube of y will equal k. We can represent the square of x as x2 and the cube of y as y3. Now, we can write the equation for inverse variation.

x2y3 = k

We are told that when x = 8, y = 8. We can substitute these values into our equation for inverse variation and then solve for k.

82(83) = k

k = 82(83)

Because this will probably be a large number, it might help just to keep it in exponent form. Let's apply the property of exponents which says that abac = ab+c.

k = 82(83) = 82+3 = 85.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 85 in for k.

x2y3 = 85

(1)2y3 = 85

y3 = 85

In order to solve this, we will have to take a cube root. Thus, it will help to rewrite 8 as the cube of 2, or 23.

y3 = (23)5

We can now apply the property of exponents that states that (ab)c = abc.

y3 = 23•5 = 215

In order to get y by itself, we will have the raise each side of the equation to the 1/3 power.

(y3)(1/3) = (215)(1/3)

Once again, let's apply the property (ab)c = abc.

y(3 • 1/3) = 2(15 • 1/3)

y = 25 = 32

The answer is 32.

Compare your answer with the correct one above

Question

varies directly as and inversely as .

$z = y ^{2}$ and $B = \sqrt{A}$ .

Which of the following is true about ?

Answer

varies directly as and inversely as , so for some constant of variation ,

$\frac{xy}{A} =K$ .

We can square both sides to obtain:

$\left (\frac{xy}{A} \right ) ^{2}=K^{2}$

$\frac{x^{2}y^{2}}{A^{2}}= K^{2}$

$z = y ^{2}$ .

$B = \sqrt{A}$ , so $B^{4} =\left ( \sqrt{A} \right )^{4} = A^{2}$ .

By substitution,

$\frac{x^{2}z}{B^{4}}= K^{2}$ .

Using $K^{2}$ as the constant of variation, we see that varies inversely as the square of and directly as the fourth power of .

Compare your answer with the correct one above

Question

The radius of the base of a cylinder is ; the height of the same cylinder is ; the cylinder has volume 1,000.

$t = \sqrt{r}$

$v = \sqrt[4]{h}$

Which of the following is a true statement?

Assume all quantities are positive.

Answer

The volume of a cylinder can be calculated from its height and the radius of its base using the formula:

$\pi r^{2} h = V$

$t = \sqrt{r}$ , so $t ^{2}= r$ ;

$v = \sqrt[4]{h}$ , so $h = v^{4}$ .

The volume is 1,000, and by substitution, using the other equations:

$\pi (t^{2})^{2} v^{4} = 1,000$

$\pi t^{4} v^{4} = 1,000$

$\frac{\pi t^{4} v^{4}}{\pi} = \frac{1,000}{\pi}$

$t^{4} v^{4} = \frac{1,000}{\pi}$

$\sqrt[4]{t^{4} v^{4} }= \sqrt[4]{\frac{1,000}{\pi}}$

$tv= \sqrt[4]{\frac{1,000}{\pi}}$

If we take $K= \sqrt[4]{\frac{1,000}{\pi}}$ as the constant of variation, we get

,

meaning that varies inversely as .

Compare your answer with the correct one above

Question

varies directly as the square of and the cube root of , and inversely as the fourth root of . Which of the following is a true statement?

Answer

varies directly as the square of and the cube root of , and inversely as the fourth root of , so, for some constant of variation ,

$\frac{s \sqrt[4]{w}}{y^{2}\sqrt[3]{t}} = K$

We take the reciprocal of both sides, then extract the square root:

$\frac{y^{2}\sqrt[3]{t}}{s \sqrt[4]{w}} =\frac{1} { K}$

$\frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} =\frac{1} { K}$

$\left ( \frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} \right )^{\frac{1}{2}} =\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y t^{\frac{1}{6}}}{s ^{\frac{1}{2}} w ^{\frac{1}{8}}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y\sqrt[6]{t}}{\sqrt{s} \cdot \sqrt[8]{w}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

Taking $\left (\frac{1} { K} \right )^{\frac{1}{2}}$ as the constant of variation, we see that varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

Compare your answer with the correct one above

Question

If varies inversely as , and when , find when .

Answer

The formula for inverse variation is as follows: $y=\frac{k}{x}$

Use the x and y values from the first part of the sentence to find k.

$14=\frac{k}{3}$

Then use that k value and the given x value to find y.

$y=\frac{42}{28}$

$y=1\frac{1}{2}$

Compare your answer with the correct one above

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