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 of rectangular prism

For a rectangular prism with length \( l \), width \( w \), height \( h \), surface area \( S = 2(lw + lh + wh) \) and volume \( V = lwh \). If the linear dimensions are scaled by a factor \( k \), the new length, width, height are \( kl \), \( kw \), \( kh \).

Step2: Find scale factor for surface area

New surface area \( S'= 2((kl)(kw)+(kl)(kh)+(kw)(kh)) = 2k^{2}(lw + lh + wh)=k^{2}S \). Given \( S' = 16S \), so \( k^{2}=16 \), then \( k = 4 \) (since scale factor is positive).

Step3: Find scale factor for volume

New volume \( V'=(kl)(kw)(kh)=k^{3}lwh = k^{3}V \). Substitute \( k = 4 \), we get \( V'=4^{3}V=64V \).

Answer:

64 (corresponding to the option with 64)