Question

the volume of this cone is 6,782.4 cubic centimeters. what is the radius of this cone?
round your answer to the nearest hundredth.
20 cm
r ≈ centimeters

Explanation:

Step1: Recall the volume formula of a cone

The volume formula of a cone is \( V=\frac{1}{3}\pi r^{2}h\), where \(V\) is the volume, \(r\) is the radius and \(h\) is the height. Given \(V = 6782.4\) cubic centimeters and \(h=20\) centimeters.

Step2: Substitute the known values into the formula

Substitute \(V\) and \(h\) into the formula: \(6782.4=\frac{1}{3}\times\pi\times r^{2}\times20\). First, simplify the right - hand side: \(\frac{1}{3}\times20\times\pi=\frac{20\pi}{3}\approx\frac{20\times3.14}{3}=\frac{62.8}{3}\). So the equation becomes \(6782.4=\frac{62.8}{3}r^{2}\).

Step3: Solve for \(r^{2}\)

Multiply both sides of the equation by \(\frac{3}{62.8}\): \(r^{2}=\frac{6782.4\times3}{62.8}\). Calculate \(6782.4\times3 = 20347.2\), then \(r^{2}=\frac{20347.2}{62.8}=324\).

Step4: Solve for \(r\)

Take the square root of both sides. Since \(r>0\), \(r=\sqrt{324} = 18.00\) (because \(\sqrt{324}=18\) and when rounded to the nearest hundredth, \(18 = 18.00\)).

Answer:

\(18.00\) centimeters