Question

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math 8
u5, l15 - volume of a cone
5.15 exit ticket
a cone with the same base but a height 3 times taller than the given cylinder exists. what is the volume of each figure? express your answers in terms of π.
image of a cylinder with height 3 and diameter 4 (or radius 2, inferred from diameter 4)

Explanation:

Step1: Find the radius and height of the cylinder

From the diagram, the diameter of the base of the cylinder is 4, so the radius \( r=\frac{4}{2} = 2\). The height of the cylinder \( h_{cylinder}=3\).

Step2: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \( V_{cylinder}=\pi r^{2}h\). Substituting \( r = 2\) and \( h=3\), we get \( V_{cylinder}=\pi\times(2)^{2}\times3=\pi\times4\times3 = 12\pi\).

Step3: Determine the height of the cone

The cone has the same base as the cylinder, so its radius \( r = 2\). The height of the cone \( h_{cone}=3\times h_{cylinder}=3\times3 = 9\).

Step4: Calculate the volume of the cone

The formula for the volume of a cone is \( V_{cone}=\frac{1}{3}\pi r^{2}h\). Substituting \( r = 2\) and \( h = 9\), we get \( V_{cone}=\frac{1}{3}\pi\times(2)^{2}\times9=\frac{1}{3}\pi\times4\times9 = 12\pi\).

Answer:

The volume of the cylinder is \( 12\pi \) and the volume of the cone is \( 12\pi \).