Question

the volume of this cone is 1,271.7 cubic feet. what is the height of this cone? round your answer to the nearest hundredth. (there is an image of a cone with radius 9 ft and a box to fill in the height h in feet)

Explanation:

Step1: Recall the volume formula for a cone

The volume \( V \) of a cone is given by the formula \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height. We know that \( V = 1271.7 \) cubic feet and \( r=9 \) feet. We need to solve for \( h \).

First, substitute the known values into the formula:
\( 1271.7=\frac{1}{3}\pi(9)^{2}h \)

Step2: Simplify the equation

Calculate \( (9)^{2}=81 \), so the equation becomes:
\( 1271.7=\frac{1}{3}\pi\times81h \)
Simplify \( \frac{1}{3}\times81 = 27 \), so:
\( 1271.7 = 27\pi h \)

Step3: Solve for \( h \)

Divide both sides of the equation by \( 27\pi \) to isolate \( h \):
\( h=\frac{1271.7}{27\pi} \)
We know that \( \pi\approx3.14159 \), so \( 27\pi\approx27\times3.14159 = 84.823 \)
Then \( h=\frac{1271.7}{84.823}\approx14.99 \) (rounded to the nearest hundredth)

Answer:

\( 14.99 \)