Question

quiz 2: surface areas of circula
a right cone has an altitude of 4 units and a base diameter of 6 units. what is its lateral area?
37.7 units²
47.1 units²
62.8 units²
75.4 units²

Explanation:

Step1: Find the radius of the base

The diameter of the base is 6 units, so the radius \( r = \frac{6}{2}=3 \) units.

Step2: Find the slant height of the cone

The altitude (height \( h \)) is 4 units. Using the Pythagorean theorem for the cone (\( l = \sqrt{r^{2}+h^{2}} \)), where \( l \) is the slant height. So \( l=\sqrt{3^{2} + 4^{2}}=\sqrt{9 + 16}=\sqrt{25} = 5 \) units.

Step3: Calculate the lateral (or curved) surface area of the cone

The formula for the lateral surface area of a cone is \( LSA=\pi rl \). Substituting \( r = 3 \) and \( l=5 \), we get \( LSA=\pi\times3\times5 = 15\pi\approx15\times3.14 = 47.1 \) units².

Answer:

47.1 units² (the option with 47.1 units²)