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

The area of the base

Total amount

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The area of the base of the cylinder can be found by first determining the volume of the cone and the cylinder separately, and then using the given total volume to find the area of the base of the cylinder.

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

Given that the height of the cone ( h = 42 ) and the radius ( r = 42 ), the volume of the cone is ( V_{\text{cone}} = \frac{1}{3} \pi \times 42^2 \times 42 ).

The volume ( V_{\text{cylinder}} ) of a cylinder is given by the formula ( V_{\text{cylinder}} = \pi r^2 h ), where ( r ) is the radius of the cylinder and ( h ) is the height of the cylinder.

Given that the height of the cylinder ( h = 1 ) and the radius ( r = 42 ), the volume of the cylinder is ( V_{\text{cylinder}} = \pi \times 42^2 \times 1 ).

The total volume ( V_{\text{total}} ) of the solid is the sum of the volumes of the cone and the cylinder, which is ( V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} ).

Substituting the given values, we get ( V_{\text{total}} = \frac{1}{3} \pi \times 42^2 \times 42 + \pi \times 42^2 \times 1 ).

Given that ( V_{\text{total}} = 480 \pi ), we can solve for the area of the base of the cylinder ( A_{\text{cylinder}} ) using the formula ( V_{\text{total}} = \pi r^2 h + \pi r^2 h_cylinder ), where ( h_cylinder ) is the height of the cylinder.

Rearranging the formula, we get ( A_{\text{cylinder}} = \frac{V_{\text{total}}}{\pi} - \pi r^2 ).

Substituting the values, we get ( A_{\text{cylinder}} = \frac{480 \pi}{\pi} - \pi \times 42^2 ).

Solving this, we find ( A_{\text{cylinder}} = 480 - 1764 ).

Thus, the area of the base of the cylinder is ( A_{\text{cylinder}} = 480 - 1764 = -1284 ) square units.

Note: The negative value indicates that there might be an error in the calculations or in the initial setup of the problem.

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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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