Question

the volume of this cone is 2,863.68 cubic yards. what is the radius of this cone? use π ≈ 3.14 and round your answer to the nearest hundredth. 19 yd (image of cone) r ≈ \boxed{} yards

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 and \( h \) is the height. We know \( V = 2863.68 \) cubic yards, \( h=19 \) yards, and \( \pi\approx3.14 \). We need to solve for \( r \).
First, we can rearrange the formula to solve for \( r^{2} \):
\[
r^{2}=\frac{3V}{\pi h}
\]

Step2: Substitute the known values into the formula

Substitute \( V = 2863.68 \), \( \pi = 3.14 \), and \( h = 19 \) into the formula for \( r^{2} \):
\[
r^{2}=\frac{3\times2863.68}{3.14\times19}
\]
First, calculate the numerator: \( 3\times2863.68 = 8591.04 \)
Then, calculate the denominator: \( 3.14\times19=59.66 \)
Now, divide the numerator by the denominator: \( r^{2}=\frac{8591.04}{59.66}\approx144 \)

Step3: Solve for \( r \)

Take the square root of both sides to find \( r \):
\[
r=\sqrt{144} = 12
\]
Wait, let's check the calculation again. Wait, maybe I made a mistake in the height? Wait, the diagram shows the height is 19? Wait, no, let's re - calculate:

Wait, \( V=\frac{1}{3}\pi r^{2}h \), so \( 3V=\pi r^{2}h \), so \( r^{2}=\frac{3V}{\pi h} \)

\( 3\times2863.68 = 8591.04 \)

\( \pi\times h=3.14\times19 = 59.66 \)

\( 8591.04\div59.66 = 144 \)

Then \( r=\sqrt{144}=12.00 \) (rounded to the nearest hundredth)

Wait, that seems correct. Let's verify with the volume formula:

\( V=\frac{1}{3}\times3.14\times12^{2}\times19=\frac{1}{3}\times3.14\times144\times19 = 3.14\times48\times19=3.14\times912 = 2863.68 \), which matches the given volume.

Answer:

\( 12.00 \)