# 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 #6 #. If the volume of the solid is #60 pi#, what is the area of the base of the cylinder?

Area of base of cylinder is

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To find the area of the base of the cylinder, we first need to find the radius of the cone and the cylinder. Since the radius of the cone is equal to the radius of the cylinder, let's denote the radius as ( r ).

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

[ \text{Volume} = \text{Volume of cone} + \text{Volume of cylinder} ]

The volume of a cone is given by the formula:

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

And the volume of a cylinder is given by:

[ V_{\text{cylinder}} = \pi r^2 h ]

Given that the height of the cone is 42 and the height of the cylinder is 6, and the total volume is ( 60\pi ), we can write the equation:

[ 60\pi = \frac{1}{3}\pi r^2 \times 42 + \pi r^2 \times 6 ]

Simplify the equation:

[ 60\pi = 14\pi r^2 + 6\pi r^2 ]

Combine like terms:

[ 60\pi = 20\pi r^2 ]

Divide both sides by ( 20\pi ):

[ r^2 = 3 ]

Take the square root of both sides:

[ r = \sqrt{3} ]

So, the radius of the base of the cylinder (and the cone) is ( \sqrt{3} ).

Now, to find the area of the base of the cylinder, we use the formula for the area of a circle:

[ A = \pi r^2 ]

Substitute ( r = \sqrt{3} ):

[ A = \pi (\sqrt{3})^2 = 3\pi ]

Therefore, the area of the base of the cylinder is ( 3\pi ) square units.

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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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