Question

A cone has radius 9 and a height 12. A frustum of this cone has height 4.

What are the radii of the bases of the frustum?
What is the slant height of the frustum?
What is the lateral area of the frustum?

(If you waste an answer it will be removed so don't try and get "Free" points.)

Answer 1

Part A)

The lower base has a radius of 9 units, and the upper base has a radius of 6 units.

Part B)

5 units.

Part C)

75π or about 235.62 square units.

Explanation:

Please refer to the figure below.

The cone has a radius of 9 units and a total height of 12 units.

A frustum of the cone has a height of 4 units.

Part A)

The lower radius of the frustum will simply be 9 units.

For the upper radius, we will use the properties of similar triangles. We will compare the smaller upper triangle to the overall larger triangle. This yields:

`\displaystyle (12)/(8)=(9)/(x)`

Solve for x. Simplify:

`\displaystyle (3)/(2)=(9)/(x)`

Cross-multiply:

`18=3x\Rightarrow x=6`

The upper base has a radius of 6 units.

Part B)

We can first find the total slant height of the entire cone. By using the Pythagorean Theorem, this yields that the total slant height is:

`SH^2=12^2+9^2`

Simplify:

`SH^2=225\Rightarrow SH=15\text{ units}`

Now, find the slant height of the upper cone:

`(SH_\text{cone})^2=8^2+6^2=100`

So:

`SH_\text{cone}=10`

Then the slant height of the frustum will be the cone subtracted from the total. Thus:

`SH_{\text{frustum}}=15-10=5\text{ units}`

Part C)

We can first find the lateral area of the entire cone. The lateral area is given by:

`LA=\pi r\ell`

The lateral area of the entire cone will be:

`LA=\pi (9)(15)=135\pi`

The lateral area of the upper cone will be:

`LA_\text{cone}=\pi(6)(10)=60\pi`

Then the lateral area of the frustum is:

`LA_\text{frustum}=135\pi-60\pi =75\pi\text{ units}^2\approx235.62\text{ units}^2`