SAT Question e6c557f9: Area and Volume answer and explanation with Desmos solution

Choice A is correct. Since all cubes are similar to each other, cube X and cube Y are similar figures. In similar figures, if lengths differ by a scale factor of $k$, then areas differ by a scale factor of $k^2$. It’s given that the edge length, in inches, of cube Y is $\frac{3}{88}$ the edge length, in inches, of cube X. It follows that the surface area, in square inches, of cube Y is ${\left(\frac{3}{88}\right)}^2$, or $\frac{9}{7,744}$, times the surface area, in square inches, of cube X. Therefore, the value of $n$ is $\frac{9}{7,744}$.

Alternate approach: The surface area of a cube is $6s^2$, where $s$ is the edge length. Let $x$ represent the edge length, in inches, of cube X. Substituting $x$ for $s$ in $6s^2$ yields $6x^2$. Therefore, the surface area, in square inches, of cube X can be expressed as $6x^2$. It’s given that the edge length, in inches, of cube Y is $\frac{3}{88}$ the edge length, in inches, of cube X. Therefore, the edge length, in inches, of cube Y can be expressed as $\left(\frac{3}{88}\right)x$, or $\frac{3x}{88}$. Substituting $\frac{3x}{88}$ for $s$ in $6s^2$ yields $6{\left(\frac{3x}{88}\right)}^2$, which is equivalent to $6\left(\frac{9x^2}{7,744}\right)$, or $\frac{27x^2}{3,872}$. Therefore, the surface area, in square inches, of cube Y can be expressed as $\frac{27x^2}{3,872}$. It’s given that the surface area, in square inches, of cube Y is $n$ times the surface area, in square inches, of cube X. It follows that $\frac{27x^2}{3,872}=n(6x^2)$. Dividing both sides of this equation by $6x^2$ yields $\frac{27x^2}{3,872(6x^2)}=n$, which is equivalent to $\frac{27x^2}{23,232x^2}=n$, or $\frac{9}{7,744}=n$. Therefore, the value of $n$ is $\frac{9}{7,744}$.