Question

a machine part consists of a half - sphere and a cylinder, as shown in the figure. the total volume of the part is π cubic inches.

Explanation:

Step1: Find radius of the shapes

The diameter of the cylinder and hemisphere is 6 inches, so the radius $r = 3$ inches.

Step2: Calculate volume of the hemisphere

The volume formula for a hemisphere is $V_{hemisphere}=\frac{2}{3}\pi r^{3}$. Substituting $r = 3$ inches, we get $V_{hemisphere}=\frac{2}{3}\pi(3)^{3}=\frac{2}{3}\pi\times27 = 18\pi$ cubic - inches.

Step3: Calculate volume of the cylinder

The volume formula for a cylinder is $V_{cylinder}=\pi r^{2}h$. Here, $r = 3$ inches and $h = 12$ inches. So $V_{cylinder}=\pi(3)^{2}\times12=\pi\times9\times12 = 108\pi$ cubic - inches.

Step4: Calculate total volume

The total volume $V = V_{hemisphere}+V_{cylinder}$. So $V=18\pi + 108\pi=126\pi$ cubic - inches.

Answer:

$126\pi$