# SAT Math : How to use the inverse variation formula

## Example Questions

### Example Question #3 : Direct And Inverse Variation

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

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 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

### Example Question #2 : Direct And Inverse Variation

varies directly as and inversely as .

and .

Which of the following is true about ?

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

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

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

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 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 ,

.

We can square both sides to obtain:

.

, so .

By substitution,

.

Using as the constant of variation, we see that varies inversely as the square of and  directly as the fourth power of .

### Example Question #6 : 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.

Which of the following is a true statement?

Assume all quantities are positive.

varies directly as the square of .

varies dorectly as the square root of .

varies inversely as the square of .

varies inversely as the square root of .

varies inversely as .

varies inversely as .

Explanation:

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

, so ;

, so .

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

If we take as the constant of variation, we get

,

meaning that varies inversely as .

### 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?

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 root 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 power 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 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 .

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 ,

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

Taking 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 .

### Example Question #2 : How To Use The Inverse Variation Formula

If  varies inversely as , and  when , find  when .