Question

prob #14
the pair of solids is similar. find the volume of each solid. round to the nearest tenth, if necessary.
volume of small solid =
ft³
volume of large solid =
ft³
need help?

Explanation:

Step1: Identify the dimensions of the small solid

The small solid (blue) is a rectangular prism with length \( l = 3 \) ft, width \( w = 3 \) ft, and height \( h = 3 \) ft. The formula for the volume of a rectangular prism is \( V = l \times w \times h \).

$$ V_{\text{small}} = 3 \times 3 \times 3 $$

Step2: Calculate the volume of the small solid

$$ V_{\text{small}} = 27 \text{ ft}^3 $$

Step3: Identify the scale factor between the solids

The height of the small solid is \( 3 \) ft and the height of the large solid (green) is \( 9 \) ft. The scale factor \( k \) is \( \frac{9}{3}=3 \). For similar solids, the ratio of volumes is \( k^3 \).

Step4: Calculate the volume of the large solid

Since the ratio of volumes is \( k^3 = 3^3 = 27 \), we multiply the volume of the small solid by \( 27 \).

$$ V_{\text{large}} = 27 \times 27 $$

$$ V_{\text{large}} = 729 \text{ ft}^3 $$

Answer:

Volume of small solid = \( 27.0 \) ft³
Volume of large solid = \( 729.0 \) ft³