While you are describing the solid, use the dimensions of the given plane region.
Solid: See solution. Volume: 24π
Practice makes perfect
Let's consider the given plane region.
We will revolve this region completely about the line y=-1.
Let's do it!
As we can see, the solid is a cone with the top of the cone cut off and with a cylinder removed from its center. To find the volume of the solid, we should find the volume of the larger cone, the volume of the smaller cone, and the volume of the cylinder using the following formulas.
c|c
Volume of a Cone&Volume of a Cylinder
V=1/3π r^2 * h & V=π r^2 * h
In the formulas r is the radius and h is the height of the solids. We will first find the height of the cone cut off the top. To do so we will use similar triangles. Let's show the triangles that we will use and label their vertices.
From here, we can conclude by the Angle-Angle Similarity Theorem that △ ABC and △ ADE are similar. Since the corresponding sides of similar triangles are proportional, we can write the following proportion.
h_s/h_s+4=1/4
Let's solve this equation for h_s.
The height of the cone cut off the top is 43. With this information, we can find the volume of the cones and the cylinder.
Solid
Radius
Height
Formula
Substitution
Volume
Larger Cone
4
4+4/3= 16/3
V=1/3π r^2 h
V=1/3π ( 4)^2 ( 16/3)
256/9π
Smaller Cone
1
4/3
V=1/3π r^2 h
V=1/3π ( 1)^2 ( 4/3)
4/9π
Cylinder
1
4
V=π r^2 h
V=π ( 1)^2 ( 4)
4π
Finally, we can find the volume of the solid by subtracting the volumes of the smaller cone and cylinder from the volume of the larger cone.
V&=256/9π-4/9π-4π
&=24π
Therefore, the volume of the solid is 24π cubic units.
