Question

question 13
compute the volume of the cone shown if given are the slant height a = 50 mi and height h = 48 mi. round your answer to two decimal places.
v = mi³
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Explanation:

Step1: Find the radius \( r \)

Using the Pythagorean theorem for the cone: \( r = \sqrt{a^2 - h^2} \). Substitute \( a = 50 \) and \( h = 48 \):
\( r = \sqrt{50^2 - 48^2} = \sqrt{2500 - 2304} = \sqrt{196} = 14 \) mi.

Step2: Calculate the volume \( V \)

The formula for the volume of a cone is \( V = \frac{1}{3}\pi r^2 h \). Substitute \( r = 14 \), \( h = 48 \):
\( V = \frac{1}{3} \pi (14)^2 (48) \).
First, compute \( 14^2 = 196 \), then \( 196 \times 48 = 9408 \).
Then, \( \frac{1}{3} \times 9408 = 3136 \).
Finally, \( V = 3136\pi \approx 3136 \times 3.14159265 \approx 9852.03 \) mi³ (rounded to two decimals).

Answer:

\( 9852.03 \)