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 260 cm^3, find the volume of the frustum.
The smaller cone is 1/3 as tall as the original cone.
So, its volume is 1/3^3 = 1/27 of the total.
That is, the frustrum has 26 times the volume of the small cone.
To find the volume of the frustum, we need to determine the volume of both the smaller cone and the original bigger cone.
Let's start by calculating the volume of the smaller cone.
The formula for the volume of a cone is given by V = (1/3) * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14159), r is the radius of the base, and h is the height of the cone.
We are given the height of the smaller cone as 8 cm. However, we need to find the radius to calculate its volume.
Since the smaller cone is cut from the bigger cone, the radius of the smaller cone's base is equal to a section of the radius of the bigger cone's base.
Let's denote the radius of the bigger cone as R and the radius of the smaller cone as r.
Given that the height of the frustum (the remaining part of the bigger cone) is 16 cm, we can use the concept of similar triangles to find the ratio of the heights and radii of the bigger and smaller cones.
The ratio of the heights is h₁/h₂ = r/R, where h₁ is the height of the smaller cone (8 cm) and h₂ is the height of the frustum (16 cm).
Therefore, r/R = 8/16 = 1/2.
Now, we have the radius of the smaller cone (r) in terms of the radius of the bigger cone (R). We can substitute this into the formula for the volume of the smaller cone and solve for its volume.
The volume of the smaller cone is V₁ = (1/3) * π * (r^2) * h₁.
Substituting r = (1/2)R and h₁ = 8 cm, we get:
V₁ = (1/3) * π * ((1/2)R)^2 * 8
= (1/3) * π * (1/4)R^2 * 8
= (1/3) * π * (R^2) * 2
= (2/3) * π * R^2
Now we have the volume of the smaller cone as (2/3) * π * R^2.
Given that the volume of the smaller cone is 260 cm^3, we can equate it to the expression we derived:
(2/3) * π * R^2 = 260
Now we can solve for R.
Dividing both sides by (2/3) * π:
R^2 = 260 / ((2/3) * π)
= (3 * 260) / (2 * π)
≈ 123.84
Taking the square root of both sides:
R ≈ √123.84
≈ 11.12 cm
Now that we have the radius of the bigger cone as approximately 11.12 cm, we can find its volume using the formula for the volume of a cone.
The volume of the bigger cone is:
V₂ = (1/3) * π * R^2 * h₂
= (1/3) * π * (11.12^2) * 16
Calculating this expression:
V₂ ≈ 2288.33 cm^3
Therefore, the volume of the frustum is approximately 2288.33 cm^3.