Question

the volume of a cone is closely related to the volume of a cylinder when both have the same base area and height. a cones volume is exactly one - third that of a cylinder with the identical dimensions mentioned. the volume of a cylinder is calculated as ( v = bh ), where ( b ) is the area of the base and ( h ) is the height. for a cone, the volume is expressed as: ( v=\frac{1}{3}bh ). a cylinder has a circular base with a radius of 6 units and a height of 19 units. if a cone has the same base area and height, what is the volume of the cone? round your answer to the nearest whole number. the volume of the cone is (square) cubic units.

Explanation:

Step1: Calculate the base area \( B \)

The base of the cylinder (and thus the cone) is a circle. The formula for the area of a circle is \( B=\pi r^{2} \). Given \( r = 6 \) units, then \( B=\pi\times6^{2}=\pi\times36 = 36\pi\) square units.

Step2: Substitute \( B \) and \( h \) into the cone - volume formula

The formula for the volume of a cone is \( V=\frac{1}{3}Bh \). We know \( B = 36\pi\) and \( h=19 \) units.
Substitute these values: \( V=\frac{1}{3}\times36\pi\times19\).
First, \( \frac{1}{3}\times36 = 12 \). So \( V=12\pi\times19\).
Then \( V = 228\pi\).
Using \( \pi\approx3.14 \), we have \( V\approx228\times3.14=715.92\).

Answer:

\( 716 \)