Question

find the volume of a right circular cone that has a height of 9.1 m and a base with a circumference of 12.1 m. round your answer to the nearest tenth of a cubic meter.

Explanation:

Step1: Find the radius from the circumference

The formula for the circumference of a circle is \( C = 2\pi r \), so we can solve for \( r \): \( r=\frac{C}{2\pi} \). Given \( C = 12.1 \) m, then \( r=\frac{12.1}{2\pi} \).

Step2: Calculate the radius

\( r=\frac{12.1}{2\pi}\approx\frac{12.1}{6.2832}\approx1.926 \) m.

Step3: Use the cone volume formula

The volume \( V \) of a right circular cone is given by \( V=\frac{1}{3}\pi r^{2}h \), where \( h = 9.1 \) m. Substitute \( r\approx1.926 \) and \( h = 9.1 \) into the formula: \( V=\frac{1}{3}\pi(1.926)^{2}(9.1) \).

Step4: Compute the volume

First, calculate \( (1.926)^{2}\approx3.709 \). Then, \( \frac{1}{3}\pi\times3.709\times9.1\approx\frac{1}{3}\times3.1416\times3.709\times9.1 \). \( 3.1416\times3.709\approx11.65 \), \( 11.65\times9.1\approx106.015 \), \( \frac{106.015}{3}\approx35.3 \) cubic meters.

Answer:

\( 35.3 \) cubic meters