The three-dimensional figure below is a cylinder with a hole in the shape of a rectangular prism going through the center of it. The radius is 10

Question

feet. Find the volume of the solid in cubic feet. Use 3.4
for pi
.

A three-dimensional cylinder is shown with a hole in the shape of a rectangular prism going through the center of it. The height of the cylinder is 15 feet. The rectangular prism-shaped-hole is 3 feet wide and 5 feet long, and 15 feet high.

Answer 1

To find the volume of the solid, we first need to calculate the volume of the cylinder and then subtract the volume of the hole (rectangular prism).

Step 1: Calculate the Volume of the Cylinder

The volume \( V \) of a cylinder can be calculated using the formula:

\[ V = \pi r^2 h \]

where \( r \) is the radius and \( h \) is the height.

Given the radius \( r = 10 \) feet and height \( h = 15 \) feet, we have:

\[ V = 3.4 \cdot (10^2) \cdot 15 \] \[ = 3.4 \cdot 100 \cdot 15 \] \[ = 3.4 \cdot 1500 \] \[ = 5100 \text{ cubic feet} \]

Step 2: Calculate the Volume of the Rectangular Prism (Hole)

The volume \( V \) of a rectangular prism can be calculated using the formula:

\[ V = l \cdot w \cdot h \]

where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Given the dimensions for the hole: length \( l = 5 \) feet, width \( w = 3 \) feet, and height \( h = 15 \) feet, we have:

\[ V = 5 \cdot 3 \cdot 15 \] \[ = 15 \cdot 15 \] \[ = 225 \text{ cubic feet} \]

Step 3: Find the Volume of the Solid

To find the volume of the solid, we subtract the volume of the hole from the volume of the cylinder:

\[ \text{Volume of Solid} = \text{Volume of Cylinder} - \text{Volume of Hole} \] \[ = 5100 - 225 \] \[ = 4875 \text{ cubic feet} \]

Thus, the volume of the solid is 4875 cubic feet.