We want to find the volume of a composite solid in the shape of a cylinder with a hemisphere hollowed out of it.
We will first find the volume of the cylinder and then the volume of the hemisphere. Then we can subtract the latter from the former to find the volume of the composite solid. Let's do it!
Volume of the Cylinder
Let's begin by finding the volume of the cylinder.
The volume of a cylinder is the product of the area of the base and the height.
V=Bh
Our cylinder has a circular base, B, and its area is the product of π times the radius squared.
V=Bh ⇒ V = π r^2 h
Let's substitute the corresponding values into the formula and find its volume.
The shape of a hemisphere is hollowed out of the cylinder.
The volume of a hemisphere is half the volume of a sphere.
V=43π r^3/2 ⇔ V = 2/3π r^3
Our hemisphere-shaped hollow has a radius of 5 inches. Let's find its volume.
To find the volume of the composite solid, we will subtract the volume of the hemisphere from the volume of the cylinder.
V_(solid)= V_(cylinder)- V_(hemisphere)
⇓
V_(solid)= 225π- 250/3π = 425/3 π
The composite solid has the volume 4253 π ≈ 445.06 in.^3.
