Question

find the approximate volume of the solid. to do the approximation, use the π button on a calculator. round your answer to the nearest tenth.
6 in
7.5 in
6 in

Explanation:

Step1: Identify the solids

The figure consists of a sphere and a cylinder. The sphere has a radius \( r = 6 \) in, and the cylinder has a radius \( r = 6 \) in and height \( h = 7.5 \) in.

Step2: Volume of the sphere

The formula for the volume of a sphere is \( V_{sphere}=\frac{4}{3}\pi r^{3} \). Substituting \( r = 6 \):

$$ V_{sphere}=\frac{4}{3}\pi(6)^{3}=\frac{4}{3}\pi\times216 = 288\pi $$

Step3: Volume of the cylinder

The formula for the volume of a cylinder is \( V_{cylinder}=\pi r^{2}h \). Substituting \( r = 6 \) and \( h = 7.5 \):

$$ V_{cylinder}=\pi(6)^{2}(7.5)=\pi\times36\times7.5 = 270\pi $$

Step4: Total volume

Add the volumes of the sphere and the cylinder:

$$ V_{total}=V_{sphere}+V_{cylinder}=288\pi + 270\pi=558\pi $$

Using a calculator with \( \pi \approx 3.14159 \):

$$ V_{total}\approx558\times3.14159\approx1752.6 $$

Answer:

The approximate volume of the solid is \( \boxed{1752.6} \) cubic inches.