select the correct answer.\nthe square of y varies directly as the cube of x. when x = 4, y = 2. which…

select the correct answer.\nthe 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?\na. $y^{2}=\frac{1}{16}x^{3}$\nb. $y=\frac{1}{2}x$\nc. $xy = 8$\nd. $x^{3}y^{2}=128$

select the correct answer.\nthe 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?\na. $y^{2}=\frac{1}{16}x^{3}$\nb. $y=\frac{1}{2}x$\nc. $xy = 8$\nd. $x^{3}y^{2}=128$

Answer

Answer:

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

Explanation:

Step1: Write direct - variation formula

If $y^{2}$ varies directly as $x^{3}$, the formula is $y^{2}=kx^{3}$, where $k$ is the constant of variation.

Step2: Find the value of $k$

Substitute $x = 4$ and $y = 2$ into $y^{2}=kx^{3}$. We get $(2)^{2}=k(4)^{3}$. So, $4 = k\times64$. Then $k=\frac{4}{64}=\frac{1}{16}$.

Step3: Write the equation

Substitute $k=\frac{1}{16}$ back into $y^{2}=kx^{3}$, we obtain $y^{2}=\frac{1}{16}x^{3}$.