A small cone of height 8 cm is cut off from a bigger cone to leave a frustum of height 16 cm. If the volume of the smaller cone is 160 cubic centimeter find the volume of the frustum
Draw a cross-section of the cone
The height of the whole original cone is 24 cm
the volumes of similar solids are proportional to the cube of their corresponding sides, so ...
volumewhole/160 = 24^3/8^3 = 27
volumewhole = 160(27) = .... cm^3
so volume of frustum = volumewhole - 160 cm^3
= .....
or, the long way:
find the radius r of the small cone which was cut off
(1/3)π(8)r^2 = 160
r = ...
let the radius of the large (original) cone be R
by ratios:
R/r = 24/8 = 3
R = 3r = ...
Now find the volume of each of the cones
frustum = volume of large cone - volume of small cone
(I got the same answer)
To find the volume of the frustum, we need to first calculate the volume of the bigger cone and then subtract the volume of the smaller cone.
Let's follow these steps:
Step 1: Find the volume of the smaller cone.
Given that the volume of the smaller cone is 160 cubic centimeters and the formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the base radius and h is the height, we can substitute the given values into the formula:
160 = (1/3) * π * r^2 * 8
Step 2: Solve for the radius, r.
First, let's simplify the equation:
160 = (8/3) * π * r^2
Now, isolate r^2:
r^2 = (160 * 3) / (8 * π)
r^2 = 180 / π
Taking the square root of both sides, we get:
r ≈ √(180 / π)
Step 3: Find the height of the bigger cone.
Given that the height of the frustum (16 cm) is equal to the height of the bigger cone minus the height of the smaller cone (8 cm), we can find the height of the bigger cone:
16 = h - 8
h = 16 + 8
h = 24 cm
Step 4: Find the volume of the bigger cone.
Using the formula V = (1/3) * π * r^2 * h, we can calculate the volume of the bigger cone:
V = (1/3) * π * (√(180 / π))^2 * 24
Step 5: Subtract the volume of the smaller cone from the volume of the bigger cone to get the volume of the frustum.
Now that we have the volumes of the smaller and bigger cones, we can find the volume of the frustum by subtracting the former from the latter:
Volume of frustum = Volume of bigger cone - Volume of smaller cone
I hope this helps you understand how to find the volume of a frustum by subtracting the volumes of the cones.