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 cm'. find the volume of the frustum.
volume of smaller cone:
(1/3)π r^2 (8) = 160
r^2 = 60/π
r = √(60/π) = 4.3702 , (stored in my calculator's memory)
Make a cross-section view of the cone, which would have a height of 24 cm
if the radius of the large cone is x
then x/24 = r/8
x = 24(√(60/π)) / 8 = appr 13.1106 , also stored in memory
volume of large cone
= (1/3) π x^2 (24) = 4320
volume of frustum = 4320 - 160 cm^3
(my calculator gave me an exact value of 4320 which I can prove if needed)
Good
To find the volume of the frustum, we need to first find the radius of both the smaller and bigger cones.
Let's assume that the radius of the smaller cone is r1 and the radius of the bigger cone is r2.
The height of the frustum (16 cm) is the difference between the height of the bigger cone (h2) and the height of the smaller cone (h1), so we have:
h2 - h1 = 16 cm
Since the height of the smaller cone (8 cm) is given, we can substitute it in the equation:
h2 - 8 cm = 16 cm
Simplifying the equation, we find:
h2 = 24 cm
The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where V is the volume, r is the radius, and h is the height.
We can find the volume of the smaller cone using the given information:
160 cm^3 = (1/3) * π * r1^2 * 8 cm
To isolate r1, we need to rearrange the equation:
r1^2 = (3 * 160 cm^3) / (8 cm * π)
Simplifying the equation, we find:
r1^2 = 60 cm^2 / π
Solving for r1, we take the square root of both sides of the equation:
r1 = √(60 cm^2 / π)
Now, we can find the radius of the bigger cone (r2) by using the similar triangles relationship between the two cones.
The ratio of the radius of the smaller cone to the radius of the bigger cone is equal to the ratio of their heights:
r1 / (r2 - r1) = h1 / (h2 - h1)
Substituting the given values, we have:
√(60 cm^2 / π) / (r2 - √(60 cm^2 / π)) = 8 cm / 16 cm
Simplifying the equation, we find:
(r2 - √(60 cm^2 / π)) / √(60 cm^2 / π) = 1/2
Now, we can solve this equation to find the value of r2.
Once we have the values of r1 and r2, we can use the formula for the volume of a frustum, which is given as:
V = (1/3) * π * (r1^2 + r2^2 + (r1 * r2)) * h
By substituting the known values, we can calculate the volume of the frustum.