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 \\(\pi\\))
\\(\square\\) \\(\text{ft}^3\\)

Explanation:

Step1: Identify the shape and formula

The pavilion 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 the volume of a half - cylinder will be $V=\frac{1}{2}\pi r^{2}h$.
First, we need to find the radius of the semi - circle. The diameter of the semi - circle is given as 24 ft, so the radius $r=\frac{24}{2}=12$ ft. The length of the half - cylinder (the height of the cylinder in the full - cylinder formula) $h = 40$ ft.

Step2: Substitute the values into the formula

Substitute $r = 12$ ft, $h = 40$ ft and $\pi=3.14$ into the formula $V=\frac{1}{2}\pi r^{2}h$.
First, calculate $r^{2}$: $r^{2}=12^{2}=144$.
Then, calculate $\pi r^{2}h$: $3.14\times144\times40 = 3.14\times5760=18086.4$.
Then, take half of that value: $V=\frac{1}{2}\times18086.4 = 9043.2$.

Answer:

9043