Question

you are installing a system of ducts as shown in the picture (not drawn to scale):
the first piece is 1.75 ft x 2 ft x 10 ft. find the volume of this piece:
the second piece is 1 ft x 1.5 ft x 12 ft. find the volume of this piece:
the third piece has a diameter of 0.5 ft, and is 8 ft long. find the volume of this piece:
what is the total volume of the system?
it takes about 6 seconds (0.1 minutes) for the air to travel from one end of the system to the
other. how many cubic feet per minute (cfm) is the fan blowing?

Explanation:

Step1: Calculate the volume of the first piece

The first piece is a rectangular prism. The volume formula for a rectangular prism is \(V = l\times w\times h\). Here, \(l = 10\) ft, \(w=2\) ft, \(h = 1.75\) ft. So \(V_1=10\times2\times1.75=35\) cubic feet.

Step2: Calculate the volume of the second piece

The second piece is also a rectangular prism. Using the formula \(V = l\times w\times h\), with \(l = 12\) ft, \(w = 1\) ft, \(h=1.5\) ft. So \(V_2=12\times1\times1.5 = 18\) cubic feet.

Step3: Calculate the volume of the third piece

The third piece is a cylinder. The volume formula for a cylinder is \(V=\pi r^{2}h\). Given \(d = 0.5\) ft (so \(r=\frac{d}{2}=0.25\) ft) and \(h = 8\) ft. Then \(V_3=\pi\times(0.25)^{2}\times8=\pi\times0.0625\times8 = 0.5\pi\approx1.57\) cubic feet.

Step4: Calculate the total volume

The total volume \(V = V_1+V_2+V_3\). Substitute the values: \(V=35 + 18+1.57=54.57\) cubic feet.

Answer:

The total volume of the system is approximately \(54.57\) cubic feet.