Question

A military tent of height 8.25 m is in the form of a right circular cylinder of base diameter 30 m and height 5.5 m surmounted by a right circular cone of same base radius. Find the length of the canvas use in making the tent, if the breadth of the canvas is 1.5 m.

Answer 1

Let the radius of the cylinder be r.
Then, $r=\frac{30}{2}=15\mathrm{m}$ which will be equal to the radius of the cone.
Now, Height of the circular cone = Height of tent − Height of cylinder = 8.25 − 5.5 = 2.75 m
Lateral height of cone (l)= $\sqrt{{15}^2+2.{75}^2}$
$$\begin{gathered} =\sqrt{225+7.5625} \\ =\sqrt{232.5625} \\ =15.25\mathrm{m} \end{gathered}$$
Curved surface area of tent = curved surface area of cylinder + curved surface area of cone
$$\begin{gathered} =2\mathrm{\pi }rh+\mathrm{\pi }rl \\ =\mathrm{\pi }r\left(2h+l\right) \\ =\frac{22}{7}\times 15\left(2\times 5.5+15.25\right) \\ =\frac{22}{7}\times 15\times 26.25 \\ =1237.5{\mathrm{m}}^2 \end{gathered}$$
Let the length of canvas be x
Now, curved surface area of tent = rectangular piece of canvas
$$\begin{gathered} \Rightarrow 1237.5=x\times 1.5 \\ \Rightarrow x=825\mathrm{m} \end{gathered}$$
Hence, the length of the canvas is 825 m