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SAT Math : How to use the inverse variation formula

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Example Questions

Example Question #8 : Direct And Inverse Variation

The square of varies inversely with the cube of . If x=8 when y=8 , then what is the value of when x=1 ?

Possible Answers:

$2^{15}$

8^5

Correct answer:

Explanation:

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 a^ba^c = a^b+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 (a^b)^c = a^bc.

y³ = 2^3•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 (a^b)^c = a^bc.

y^{(3 • 1/3)}= 2^{(15 • 1/3)}

y = 2⁵ = 32

The answer is 32.

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Example Question #1 : Direct And Inverse Variation

varies directly as and inversely as .

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

Which of the following is true about ?

Possible Answers:

varies inversely as the square root of and directly as the fourth root of .

varies directly as the square root of and inversely as the fourth root of .

varies directly as both the square root of and the fourth root of .

varies inversely as the square of and directly as the fourth power of .

varies directly as the square of and inversely as the fourth power of .

Correct answer:

varies inversely as the square of and directly as the fourth power of .

Explanation:

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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Example Question #1 : Direct And Inverse Variation

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.

Possible Answers:

varies directly as the square of .

varies inversely as .

varies dorectly as the square root of .

varies inversely as the square of .

varies inversely as the square root of .

Correct answer:

varies inversely as .

Explanation:

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

tv=K ,

meaning that varies inversely as .

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Example Question #1 : How To Use The Inverse Variation Formula

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?

Possible Answers:

varies directly as the square root of , and inversely as the eighth root of and the sixth root of .

varies directly as the square root of and the eighth power of , and inversely as the sixth root of .

varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

varies directly as the square root of , and inversely as the eighth power of and the sixth power of .

varies directly as the square root of and the eighth root of , and inversely as the sixth power of .

Correct answer:

varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

Explanation:

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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Example Question #1 : How To Use The Inverse Variation Formula

If varies inversely as , and y=14 when x=3 , find when x=28 .

Possible Answers:

$37 \frac{1}{3}$

$1\frac{1}{2}$

$1 \frac{5}{7}$

$\frac{4}{7}$

Correct answer:

$1\frac{1}{2}$

Explanation:

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.

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

k=42

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

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

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

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SAT Math : How to use the inverse variation formula

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #8 : Direct And Inverse Variation

Example Question #1 : Direct And Inverse Variation

Example Question #1 : Direct And Inverse Variation

Example Question #1 : How To Use The Inverse Variation Formula

Example Question #1 : How To Use The Inverse Variation Formula

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