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## Example Questions

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

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

**Possible Answers:**

**Correct 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 x^{2} and the cube of y as y^{3}. Now, we can write the equation for inverse variation.

x^{2}y^{3} = 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.

8^{2}(8^{3}) = k

k = 8^{2}(8^{3})

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^{b}a^{c} = a^{b+c}.

k = 8^{2}(8^{3}) = 8^{2+3} = 8^{5}.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 8^{5} in for k.

x^{2}y^{3} = 8^{5}

(1)^{2}y^{3} = 8^{5}

y^{3} = 8^{5}

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 2^{3}.

y^{3} = (2^{3})^{5}

We can now apply the property of exponents that states that (a^{b})^{c} = a^{bc}.

y^{3} = 2^{3•5} = 2^{15}

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

(y^{3})^{(1/3)} = (2^{15})^{(1/3)}

Once again, let's apply the property (a^{b})^{c} = a^{bc}.

y^{(3 • 1/3) }= 2^{(15 • 1/3)}

y = 2^{5} = 32

The answer is 32.

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

varies directly as and inversely as .

and .

Which of the following is true about ?

**Possible Answers:**

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 .

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 root of and directly as the fourth root of .

**Correct answer:**

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

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 #3 : 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.

**Possible Answers:**

varies inversely as .

varies inversely as the square of .

varies inversely as the square root of .

varies directly as the square of .

varies dorectly as the square root of .

**Correct answer:**

varies inversely as .

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?

**Possible Answers:**

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 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 .

**Correct answer:**

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

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 .

**Possible Answers:**

**Correct answer:**

The formula for inverse variation is as follows:

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

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

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