How to use the inverse variation formula - SAT Math
Card 0 of 5
The square of 
varies inversely with the cube of 
. If 
when 
, then what is the value of 
when 
?
The square of
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.
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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varies directly as 
and inversely as 
.
and 
.
Which of the following is true about 
?
Which of the following is true about
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 
.
We can square both sides to obtain:
By substitution,
Using
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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.
The radius of the base of a cylinder is
Which of the following is a true statement?
Assume all quantities are positive.
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 
.
The volume of a cylinder can be calculated from its height and the radius of its base using the formula:
The volume is 1,000, and by substitution, using the other equations:
If we take
meaning that
Compare your answer with the correct one above
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 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 
.
We take the reciprocal of both sides, then extract the square root:
Taking
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If 
varies inversely as 
, and 
when 
, find 
when 
.
If
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.
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.
Compare your answer with the correct one above