Question

a cone has a slant height of 10 centimeters and a lateral area of 60π square centimeters. what is the volume of a sphere with a radius equal to that of the cone?
a 102π cm³
b 144π cm³
c 288π cm³
d 1,333π cm³
○ a
○ d
○ b
○ c

Explanation:

Step1: Find the radius of the cone

The lateral (or curved) surface area of a cone is given by the formula \( L = \pi r l \), where \( r \) is the radius of the base of the cone and \( l \) is the slant height. We know that \( L = 60\pi \) and \( l = 10 \). Plugging these values into the formula:

$$ 60\pi=\pi\times r\times10 $$

Divide both sides by \( 10\pi \):

$$ r = \frac{60\pi}{10\pi}=6 $$

So the radius of the cone (and thus the radius of the sphere) is \( r = 6 \) centimeters.

Step2: Calculate the volume of the sphere

The volume \( V \) of a sphere is given by the formula \( V=\frac{4}{3}\pi r^{3} \). We know \( r = 6 \), so we substitute this into the formula:

$$ V=\frac{4}{3}\pi\times(6)^{3} $$

First, calculate \( 6^{3}=6\times6\times6 = 216 \). Then:

$$ V=\frac{4}{3}\pi\times216 $$

Simplify \( \frac{4}{3}\times216 \): \( 216\div3 = 72 \), and \( 72\times4 = 288 \). So:

$$ V = 288\pi $$

Answer:

C. \( 288\pi \text{ cm}^3 \)