The volume of the frustum is a difference of the volume of the larger cone and the volume of the smaller cone.
A
Yes, see solution.
B
Smaller Cone: 7.07 in.^3 Bigger Cone: 56.55 in.^3
C
1/8
D
49.48 in.^3
Practice makes perfect
A frustum is the solid left after a cone is cut by a plane parallel to its base and the top cone is removed. We can see a frustum in the following graph. We want to decide if the smaller cone that is removed is similar to the original one.
To answer this question, we should check if their lengths — the radii of the bases and the heights — are proportional. We can read these lengths from the graph.
c|c
Top Cone & Original Cone
height: 3 in. & height: 6 in.
radius: 1.5 in. & radius: 3 in.
We see that the dimensions of the original cone are twice the dimensions of the top cone.
c|c
Top Cone & Original Cone
height: 3in. & height: 2( 3) =6in.
radius: 1.5in. & radius: 2( 1.5) =3 in.
Since the lengths are proportional, the cones are similar.
We are asked to find the volumes of the top cone and the original cone.
For that, let's remember the formula for the volume of a cone. The volume V of a cone with radius r and height h can be written as follows.
V = 1/3π r^2h
We can now substitute the dimensions of each cone into the formula and evaluate.
c|c
Top Cone & Original Cone
height: 3 in. & height: 6 in.
radius: 1.5 in. & radius: 3 in.
Let's start with the top cone.
We found that about 56.55 in.^3 is the volume of the original cone.
We want to find the ratio of the volume of the smaller cone V_1 to the volume of the larger cone V_2. We can use the values found in Part B of the exercise.
V_1 & = 7.07 in.^3
V_2 &= 56.55 in.^3
Calculating this ratio means dividing V_1 by V_2. Let's do it then!
