# 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 #42 # and the height of the cylinder is #1 #. If the volume of the solid is #320 pi#, what is the area of the base of the cylinder?

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Let the radius of the cone and cylinder be ( r ).

The volume of the solid is the sum of the volumes of the cone and cylinder:

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

Given that the height of the cone is 42 and its radius is ( r ), the volume of the cone is:

[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}} = \frac{1}{3} \pi r^2 \times 42 ]

Given that the height of the cylinder is 1, the volume of the cylinder is:

[ V_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}} = \pi r^2 \times 1 ]

Given that the total volume of the solid is 320π, we can write the equation as:

[ 320\pi = \frac{1}{3} \pi r^2 \times 42 + \pi r^2 ]

Solving for ( r ) gives:

[ 320\pi = 14\pi r^2 + \pi r^2 ]

[ 320\pi = 15\pi r^2 ]

[ r^2 = \frac{320\pi}{15\pi} ]

[ r^2 = \frac{64}{3} ]

[ r = \sqrt{\frac{64}{3}} ]

[ r = \frac{8}{\sqrt{3}} ]

The area of the base of the cylinder is ( \pi r^2 = \pi \left(\frac{8}{\sqrt{3}}\right)^2 ).

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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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