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

The area of the base is

Total amount

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Let's denote the radius of both the cone and the cylinder as ( r ). Given that the height of the cone is 21 and the height of the cylinder is 2, we can set up the volume equation for the solid:

[ \text{Volume of cone} + \text{Volume of cylinder} = 90\pi ]

The volume of a cone is (\frac{1}{3}\pi r^2 h), where ( r ) is the radius and ( h ) is the height. Similarly, the volume of a cylinder is (\pi r^2 h). Substituting the given values:

[ \frac{1}{3}\pi r^2 \times 21 + \pi r^2 \times 2 = 90\pi ]

Simplifying and solving for ( r ), we get:

[ 7\pi r^2 + 2\pi r^2 = 90\pi ] [ 9\pi r^2 = 90\pi ] [ r^2 = \frac{90\pi}{9\pi} ] [ r^2 = 10 ]

Taking the square root of both sides to find ( r ):

[ r = \sqrt{10} ]

The area of the base of the cylinder is given by ( \pi r^2 ), so substituting ( r = \sqrt{10} ):

[ \text{Area of base} = \pi (\sqrt{10})^2 = 10\pi ]

Therefore, the area of the base of the cylinder is ( 10\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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- How do you use Heron's formula to find the area of a triangle with sides of lengths #23 #, #21 #, and #22 #?

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