Volume of a Composite Geometric Shape
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A sand castle built in the shape of a cylinder and a right cone is constructed. The internal measurements of the sand castle are represented in the figure above. Of the following, which is closest to the volume of the sand castle, in cubic feet?
This question includes visual content: The image displays a composite 3D object consisting of a cylinder at the base with a cone stacked on top. The cylinder has a height of 200 ft and a radius of 20 ft. The cone sits on top of the cylinder with a height of 300 ft and shares the same 20 ft radius at its base. Labels indicate the dimensions clearly.
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Step by Step Written Solution
In this problem, we need to find the total volume of a sand castle that is shaped like a cylinder with a cone on top. Let us break down the dimensions from the image first.
Problem Analysis
Looking at the figure, we see a cylindrical base. The radius of this base is given as twenty feet. The height of the cylinder is two hundred feet.
Above the cylinder, there is a right cone. The cone shares the same radius of twenty feet, and its vertical height is labeled as three hundred feet.
The total volume is the sum of the volume of the cylinder and the volume of the cone. Let us write down the formula for the total volume.
Volume Formulas
Now, let's substitute our known values into the total volume equation.
First, let's calculate the volume of the cylindrical part. Twenty squared is four hundred. Multiplying four hundred by two hundred gives eighty thousand. So, the cylinder volume is eighty thousand pi.
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