Question

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

Answer 1

Answer:

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.

Step-by-step 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:

$\displaystyle \frac{12}{8}=\frac{9}{x}$

Solve for x. Simplify:

$\displaystyle \frac{3}{2}=\frac{9}{x}$

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:

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