SAT Question 502d9690: Area and Volume answer and explanation with Desmos solution

Choice C is correct. It's given that rectangle $ABCD$ is similar to rectangle $EFGH$. Therefore, if the length of each side of rectangle $ABCD$ is $k$ times the length of the corresponding side of rectangle $EFGH$, then the area of rectangle $ABCD$ is $k^2$ times the area of rectangle $EFGH$. It's given that the area of rectangle $ABCD$ is $648$ square inches and the area of rectangle $EFGH$ is $72$ square inches. It follows that $k^2=\frac{648}{72}$, or $k^2=9$. Taking the square root of each side of this equation yields $k=\sqrt{9}$, or $k=3$. It follows that the length of each side of rectangle $ABCD$ is $3$ times the length of the corresponding side of rectangle $EFGH$. It's given that the length of the longest side of rectangle $ABCD$ is $36$ inches. Therefore, $36$ inches is $3$ times the length of the longest side of rectangle $EFGH$, and the longest side of rectangle $EFGH$ is equal to $\frac{36}{3}$, or $12$, inches.