A solid consists of a cone on top of a cylinder with a radius equal to the cone. The height of the cone is #3 # and the height of the cylinder is #5 #. If the volume of the solid is #25 pi#, what is the area of the base of cylinder?

Let V be the volume of the solid figure; let V1, V2, and h1, h2, and h1 be the height of the cone and cylinder, respectively; and let h2 be the height of the cylinder.

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Given that the cone's radius and the cylinder's radius are equal, B1=B2=B

To solve for the area of the base of the cylinder, you need to understand the relationship between the volume of the solid and the volumes of the cone and the cylinder.

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

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

Given that the volume of the solid is (25\pi) and the height of the cone and cylinder are 3 and 5 respectively, you can express the volumes of the cone and cylinder using their respective formulas:

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

Where ( r ) is the radius and ( h ) is the height of the cone or cylinder.

Since the radius of the cylinder is equal to the radius of the cone, let's denote it as ( r ). Given that the height of the cone is 3 and the height of the cylinder is 5, and the volume of the solid is (25\pi), we can set up the equation:

[ 25\pi = \frac{1}{3} \pi r^2 (3) + \pi r^2 (5) ]

Solve this equation to find the value of ( r ). Once you find the value of ( r ), you can calculate the area of the base of the cylinder using the formula:

[ \text{Area of base of cylinder} = \pi r^2 ]