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Example Questions
Example Question #1 : How To Use The Inverse Variation Formula
The square of varies inversely with the cube of . If when , then what is the value of when ?
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.
Example Question #2 : How To Use The Inverse Variation Formula
varies directly as and inversely as .
and .
Which of the following is true about ?
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 .
varies directly as the square of and inversely as the fourth power 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 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 #1 : How To Use The Inverse Variation Formula
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 inversely as the square root of .
varies inversely as the square of .
varies inversely as .
varies directly as the square of .
varies dorectly as the square root of .
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 #3 : 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 the eighth root of , and inversely as the sixth power 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 power of , and inversely as the sixth root 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 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 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 : New Sat Math Calculator
If varies inversely as , and when , find when .
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.