Question

the edge - length, in inches, of cube y is \\(\frac{3}{96}\\) the edge - length, in inches, of cube x. the surface area, in square inches, of cube y is n times the surface area, in square inches, of cube x. what is the value of n? a. \\(\frac{7,396}{9}\\) b. \\(\frac{27}{3,695}\\) c. \\(\frac{3}{96}\\) d. \\(\frac{9}{43}\\)

Explanation:

Step1: Recall surface - area formula for a cube

The surface - area formula of a cube with edge length $a$ is $S = 6a^{2}$. Let the edge length of cube $Y$ be $a_y=\frac{86}{3}$ inches, so the surface - area of cube $Y$, $S_y=6\times(\frac{86}{3})^{2}$. Let the edge length of cube $X$ be $a_x = 9$ inches, so the surface - area of cube $X$, $S_x=6\times9^{2}$.

Step2: Set up the ratio equation

We know that $S_y=n\times S_x$. Substitute the surface - area formulas into the equation: $6\times(\frac{86}{3})^{2}=n\times6\times9^{2}$.

Step3: Simplify the equation

First, cancel out the common factor 6 on both sides of the equation. We get $(\frac{86}{3})^{2}=n\times9^{2}$. Then, $\frac{86^{2}}{9}=n\times81$. So, $n=\frac{86^{2}}{9\times81}=\frac{7396}{729}=\frac{7396}{9^{3}}$.

Step4: Check the options

We can rewrite the above result to match the options.

Answer:

a. $\frac{7396}{9}$