Question

the volume of this cone is 25.12 cubic centimeters. what is the height of the cone? round your answer to the nearest hundredth. (there is an image of a cone with radius 2 cm and a box to fill in the height h in centimeters)

Explanation:

Step1: Recall the volume formula of 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 of the base and \( h \) is the height of the cone. We know that \( V = 25.12\space cm^{3}\) and \( r=2\space cm \). We need to solve for \( h \).

Step2: Substitute the known values into the formula

Substitute \( V = 25.12\), \( r = 2\) into the formula \( V=\frac{1}{3}\pi r^{2}h \):
\[
25.12=\frac{1}{3}\times\pi\times(2)^{2}\times h
\]
First, calculate \( (2)^{2}=4 \), so the equation becomes:
\[
25.12=\frac{1}{3}\times\pi\times4\times h
\]
\[
25.12=\frac{4\pi}{3}h
\]

Step3: Solve for \( h \)

We can re - arrange the formula to solve for \( h \). Multiply both sides of the equation by \( 3 \) to get rid of the denominator on the right - hand side:
\[
25.12\times3 = 4\pi h
\]
\[
75.36=4\pi h
\]
Then divide both sides by \( 4\pi \) to solve for \( h \). We know that \( \pi\approx3.14 \), so \( 4\pi=4\times3.14 = 12.56 \)
\[
h=\frac{75.36}{4\pi}=\frac{75.36}{12.56}
\]
\[
h = 6
\]

Answer:

The height of the cone is \( 6.00 \) centimeters (rounded to the nearest hundredth).