Question

The cone shown has a diameter of 18 meters and a slant height of 15 meters. Which choice is closest to the lateral surface area? Use 3.14 to approximate pi.

Answer 1

If you unfurl the lateral "face" of a cone, you get the sector of a circle whose central angle subtends an arc with length $L$ equal to the circumference of the cone's base, and a radius equal to the slant height $s$ of the cone.

Since the diameter of the base is 18m, the circumference of the base is about $3.14\times18=56.52$ meters, so this is the value of $L$.

You're given a slant height of $s=15$ meters.

Now, the following proportional relation holds for any circle:

$\dfrac{\text{area of sector}}{\text{area of circle}}=\dfrac{\text{arc length of sector}}{\text{circumference of circle}}$

This translates to

$\dfrac A{\pi\times s9^2}=\dfrac s{2\pi s\times9}$

(remember, we're talking about a sector of a circle with radius $s$ and arc length $18\pi$). Solving for $A$:

$\dfrac A{81\times3.14}=\dfrac{15}{18\times15\times3.14}$
$A=\dfrac92=4.5$

So the area of the lateral face is approximately 4.5 square meters.

Answer 2

Its 424 square meters