Question

find the volume of the composite solid. round your answer to the nearest hundredth.

Explanation:

Step1: Identify Composite Shapes

The composite solid consists of a cube and a hemisphere. The cube has side length \( s = 4 \) in. The hemisphere has a diameter equal to the side of the cube, so radius \( r=\frac{4}{2}=2 \) in.

Step2: Volume of Cube

The formula for the volume of a cube is \( V_{\text{cube}} = s^3 \). Substituting \( s = 4 \):
\( V_{\text{cube}} = 4^3 = 64 \) cubic inches.

Step3: Volume of Hemisphere

The formula for the volume of a hemisphere is \( V_{\text{hemisphere}}=\frac{2}{3}\pi r^3 \). Substituting \( r = 2 \):
\( V_{\text{hemisphere}}=\frac{2}{3}\pi(2)^3=\frac{2}{3}\pi(8)=\frac{16}{3}\pi \approx \frac{16}{3}(3.1416)\approx 16.7552 \) cubic inches.

Step4: Total Volume

Add the volumes of the cube and hemisphere:
\( V_{\text{total}} = V_{\text{cube}} + V_{\text{hemisphere}} = 64 + 16.7552 \approx 80.7552 \). Rounding to the nearest hundredth: \( 80.76 \).

Answer:

The volume of the composite solid is approximately \(\boldsymbol{80.76}\) cubic inches.