Question

the surface area of a rectangular prism is increased by a factor of 16. by what factor is the volume of the figure increased? 4 8 32 64

Explanation:

Step1: Recall surface - area and volume relationships

For similar solids, if the ratio of surface - areas of two similar solids is $S_1:S_2 = k^2$, and the ratio of their volumes is $V_1:V_2=k^3$. Given that the surface area is increased by a factor of 16, so $k^2 = 16$.

Step2: Solve for the scale factor $k$

If $k^2=16$, then $k=\sqrt{16}=4$ (we take the positive value since $k$ represents a scale - factor of lengths).

Step3: Find the factor of volume increase

Since the ratio of volumes of two similar solids is $k^3$, and $k = 4$, then the factor by which the volume is increased is $k^3=4^3 = 64$.

Answer:

64