Question

posttest: rational relationships
22
select the correct answer.
the square of y varies directly as the cube of x. when x = 4, y = 2. which equation can be used to find other combinations of x and y?
a. $y^2 = \frac{1}{16}x^3$
b. $y = \frac{1}{2}x$
c. $xy = 8$
d. $x^3y^2 = 128$

Explanation:

Step1: Recall direct variation formula

Direct variation: If \( y^2 \) varies directly as \( x^3 \), then \( y^2 = kx^3 \), where \( k \) is the constant of variation.

Step2: Find the constant \( k \)

Given \( x = 4 \), \( y = 2 \). Substitute into \( y^2 = kx^3 \):
\( 2^2 = k(4^3) \)
\( 4 = k(64) \)
Solve for \( k \): \( k=\frac{4}{64}=\frac{1}{16} \)

Step3: Write the equation

Substitute \( k = \frac{1}{16} \) into \( y^2 = kx^3 \), we get \( y^2=\frac{1}{16}x^3 \)

Answer:

A. \( y^{2}=\frac{1}{16}x^{3} \)