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The square of x varies inversely with the cube of y. If x = 8 when y = 8, then what is the value of y when x = 1?
A. 85
B. 32
C. 64
D. 2
E. 215
Correct Answer: B
Explanation/Reference:
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 x 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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