# 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 #42 pi#, what is the area of the base of the cylinder?

The area is

Total amount

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Let ( r ) represent the radius of both the cone and the cylinder.

The volume ( V ) of the solid is given by the sum of the volumes of the cone and the cylinder:

[ V = V_{\text{cone}} + V_{\text{cylinder}} ]

[ V = \frac{1}{3}\pi r^2h_{\text{cone}} + \pi r^2h_{\text{cylinder}} ]

Given ( h_{\text{cone}} = 21 ) and ( h_{\text{cylinder}} = 2 ), and ( V = 42\pi ), we can substitute these values into the equation:

[ 42\pi = \frac{1}{3}\pi r^2(21) + \pi r^2(2) ]

Solve for ( r ):

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

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

[ r^2 = \frac{42\pi}{9\pi} ]

[ r^2 = \frac{14}{3} ]

[ r = \sqrt{\frac{14}{3}} ]

The area of the base of the cylinder ( A_{\text{cylinder}} ) is given by ( A_{\text{cylinder}} = \pi r^2 ). Substitute the value of ( r ):

[ A_{\text{cylinder}} = \pi \left( \sqrt{\frac{14}{3}} \right)^2 ]

[ A_{\text{cylinder}} = \pi \left( \frac{14}{3} \right) ]

[ A_{\text{cylinder}} = \frac{14\pi}{3} ]

So, the area of the base of the cylinder is ( \frac{14\pi}{3} ).

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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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