Question

find the lateral surface area and surface area of a cone with a slant height of 12 in and a base diameter of 8 in. use the value 3.14 for π, and do not do any rounding. be sure to include the correct unit in your answers. (a) lateral surface area: (b) surface area:

Explanation:

Step1: Find the radius of the base

The diameter of the base is 8 in, so the radius \( r = \frac{8}{2} = 4 \) in.

Step2: Calculate the lateral surface area of the cone

The formula for the lateral (or curved) surface area of a cone is \( \pi r l \), where \( l \) is the slant height. Substituting \( r = 4 \) in, \( l = 12 \) in, and \( \pi = 3.14 \):
\( \text{Lateral Surface Area} = 3.14 \times 4 \times 12 \)
\( = 3.14 \times 48 \)
\( = 150.72 \) square inches.

Step3: Calculate the surface area of the cone

The surface area of a cone is the sum of the lateral surface area and the area of the base. The area of the base (a circle) is \( \pi r^2 \). First, find the area of the base:
\( \text{Base Area} = 3.14 \times 4^2 = 3.14 \times 16 = 50.24 \) square inches.
Then, the surface area is the lateral surface area plus the base area:
\( \text{Surface Area} = 150.72 + 50.24 = 200.96 \) square inches.

Answer:

(a) lateral surface area: \( 150.72 \) square inches
(b) surface area: \( 200.96 \) square inches