Question

a pavilion is made by stretching fabric over semi - circle supports.
what is the volume of air inside the pavilion to the nearest cubic foot?
(use 3.14 to approximate π)
□ ft³

Explanation:

Step1: Identify the shape

The pavilion's air volume is a half - cylinder (semi - circular cross - section along the length). The formula for the volume of a full cylinder is \(V = \pi r^{2}h\), so for a half - cylinder, the volume formula is \(V=\frac{1}{2}\pi r^{2}l\) (where \(l\) is the length of the cylinder, and \(r\) is the radius of the semi - circle).

From the diagram, the diameter of the semi - circle is \(d = 24\) ft, so the radius \(r=\frac{d}{2}=\frac{24}{2}=12\) ft. The length of the half - cylinder \(l = 42\) ft.

Step2: Substitute values into the formula

We know that \(\pi\approx3.14\), \(r = 12\) ft, and \(l = 42\) ft. Substitute these values into the formula for the volume of the half - cylinder:

\(V=\frac{1}{2}\times3.14\times(12)^{2}\times42\)

First, calculate \((12)^{2}=144\). Then:

\(V=\frac{1}{2}\times3.14\times144\times42\)

\(\frac{1}{2}\times144 = 72\), so:

\(V=3.14\times72\times42\)

Calculate \(3.14\times72 = 226.08\), then:

\(V = 226.08\times42\)

\(226.08\times42=(200 + 26+0.08)\times42=200\times42+26\times42 + 0.08\times42=8400+1092+3.36 = 9495.36\)

Answer:

\(9495\) (rounded to the nearest cubic foot)