Question

homework 3: volume of composite figures
this is a 2 - page document!
directions: find the volume of each figure. round to the nearest hundredth when necessary.
1
0.5 ft
4 ft
1 ft
6 ft
2 ft
2
4 in.
4 in.
5 in.
5 in.
5 in.
3
4 m
6 m
2 m
8 m
4
4 ft
3 ft
10 ft
4 ft
5
6 in.
10 in.
6 in.
8 in.

Explanation:

1.

Step1: Split the composite figure into two rectangular - prisms

The first rectangular - prism has dimensions \(l_1 = 6\) ft, \(w_1 = 2\) ft, \(h_1 = 1\) ft. The second rectangular - prism has dimensions \(l_2 = 4\) ft, \(w_2 = 2\) ft, \(h_2 = 1 - 0.5=0.5\) ft.
The volume of a rectangular - prism is \(V = l\times w\times h\).
For the first prism: \(V_1=l_1\times w_1\times h_1=6\times2\times1 = 12\) cubic feet.
For the second prism: \(V_2=l_2\times w_2\times h_2=4\times2\times0.5 = 4\) cubic feet.

Step2: Calculate the total volume

The total volume \(V = V_1+V_2\).
\(V=12 + 4=16\) cubic feet.

Step1: Calculate the volume of the cube

The volume of a cube with side length \(s = 5\) inches is \(V_{cube}=s^3\).
\(V_{cube}=5^3=125\) cubic inches.

Step2: Calculate the volume of the cylinder

The volume of a cylinder is \(V_{cylinder}=\pi r^2h\), where \(r=\frac{4}{2}=2\) inches and \(h = 4\) inches.
\(V_{cylinder}=\pi\times2^2\times4=16\pi\approx16\times3.14 = 50.24\) cubic inches.

Step3: Calculate the total volume

The total volume \(V = V_{cube}+V_{cylinder}\).
\(V=125 + 50.24=175.24\) cubic inches.

Step1: Calculate the volume of the first cone

The volume of a cone is \(V=\frac{1}{3}\pi r^2h\). For the first cone with \(r_1 = 4\) m and \(h_1 = 8\) m, \(V_1=\frac{1}{3}\pi r_1^2h_1=\frac{1}{3}\pi\times4^2\times8=\frac{128\pi}{3}\) cubic meters.

Step2: Calculate the volume of the second cone

For the second cone with \(r_2 = 2\) m and \(h_2 = 6\) m, \(V_2=\frac{1}{3}\pi r_2^2h_2=\frac{1}{3}\pi\times2^2\times6 = 8\pi\) cubic meters.

Step3: Calculate the total volume

The total volume \(V = V_1+V_2\).
\(V=\frac{128\pi}{3}+8\pi=\frac{128\pi + 24\pi}{3}=\frac{152\pi}{3}\approx\frac{152\times3.14}{3}\approx158.85\) cubic meters.

Answer:

16 cubic feet

2.