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 canvas used in making the tent, if the breadth of the canvas is 1.5 m. [CBSE 2012]

Answer 1

We have,
the radii of bases of the cone and cylinder, r = 15 m,
the height of the cylinder, h = 5.5 m,
the height of the tent = 8.25 m
Also, the height of the cone, H = $8.25-5.5=2.75\mathrm{m}$

$$\begin{gathered} \mathrm{The}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone},l=\sqrt{r^2+H^2} \\ =\sqrt{{15}^2+2.{75}^2} \\ =\sqrt{225+7.5625} \\ =\sqrt{232.5625} \\ =15.25\mathrm{m} \end{gathered}$$

$$\begin{gathered} \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{canvas}\mathrm{used}\mathrm{in}\mathrm{making}\mathrm{the}\mathrm{tent}=\mathrm{CSA}\mathrm{of}\mathrm{cylinder}+\mathrm{CSA}\mathrm{of}\mathrm{cone} \\ =2\mathrm{\pi }rh+\mathrm{\pi }rl \\ =\mathrm{\pi }r\left(2h+l\right) \\ =\frac{22}{7}\times 15\left(2\times 5.5+15.25\right) \\ =\frac{22}{7}\times 15\left(11+15.25\right) \\ =\frac{22}{7}\times 15\times 26.25 \\ =1237.5{\mathrm{m}}^2 \end{gathered}$$

As, the width of the canvas = 1.5 m

$\mathrm{So},\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{canvas}=\frac{237.5}{1.5}=825\mathrm{m}$

Hence, the length of the tent used for making the tent is 825 m.