Question

a machine part consists of a half - sphere and a cylinder, as shown in the figure. the total volume of the part is blank π cubic inches. type the correct answer in the box. reset next

Explanation:

Step1: Calculate radius of shapes

The diameter of the half - sphere and the cylinder is 6 inches, so the radius $r = \frac{6}{2}=3$ inches.

Step2: Calculate volume of the half - sphere

The volume formula for a sphere is $V_{sphere}=\frac{4}{3}\pi r^{3}$. For a half - sphere, $V_{half - sphere}=\frac{1}{2}\times\frac{4}{3}\pi r^{3}=\frac{2}{3}\pi r^{3}$. Substituting $r = 3$ inches, we get $V_{half - sphere}=\frac{2}{3}\pi\times3^{3}=\frac{2}{3}\pi\times27 = 18\pi$ 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 $V_{cylinder}=\pi\times3^{2}\times12=\pi\times9\times12 = 108\pi$ cubic inches.

Step4: Calculate total volume

The total volume of the machine part $V = V_{half - sphere}+V_{cylinder}$. So $V=18\pi + 108\pi=126\pi$ cubic inches.

Answer:

$126\pi$