# A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #21 # and the height of the cylinder is #2 #. If the volume of the solid is #72 pi#, what is the area of the base of the cylinder?

The area of the base

Total amount

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

The volume ( V_c ) of the cone with radius ( r ) and height ( 21 ) is ( V_c = \frac{1}{3}\pi r^2 \times 21 ).

The volume ( V_{cy} ) of the cylinder with radius ( r ) and height ( 2 ) is ( V_{cy} = \pi r^2 \times 2 ).

Given that the total volume of the solid is ( 72\pi ), we can set up the equation:

[ V_c + V_{cy} = 72\pi ]

[ \frac{1}{3}\pi r^2 \times 21 + \pi r^2 \times 2 = 72\pi ]

[ 7\pi r^2 + 2\pi r^2 = 72\pi ]

[ 9\pi r^2 = 72\pi ]

[ r^2 = \frac{72\pi}{9\pi} ]

[ r^2 = 8 ]

[ r = \sqrt{8} ]

The area ( A ) of the base of the cylinder is given by ( A = \pi r^2 ).

Substituting ( r = \sqrt{8} ), we get:

[ A = \pi (\sqrt{8})^2 ]

[ A = 8\pi ]

So, the area of the base of the cylinder is ( 8\pi ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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