Question

A 5-m-wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of ₹25 per metre. [CBSE 2014]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone},h=24\mathrm{m},\mathrm{the}\mathrm{base}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{cone},d=14\mathrm{m} \\ \mathrm{Also},\mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cone},r=\frac{d}{2}=\frac{14}{2}=7\mathrm{m} \\ \mathrm{The}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone},l=\sqrt{h^2+r^2} \\ =\sqrt{{24}^2+7^2} \\ =\sqrt{576+49} \\ =\sqrt{625} \\ =25\mathrm{m} \\ \mathrm{The}\mathrm{curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{tent}=\mathrm{\pi }rl \\ =\frac{22}{7}\times 7\times 25 \\ =550{\mathrm{m}}^2 \\ \Rightarrow \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{cloth}\mathrm{required}\mathrm{to}\mathrm{make}\mathrm{the}\mathrm{tent}=550{\mathrm{m}}^2 \\ \Rightarrow \mathrm{The}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{cloth}=\frac{550}{5}=110\mathrm{m} \\ \mathrm{So},\mathrm{the}\mathrm{cost}\mathrm{of}\mathrm{cloth}\mathrm{used}=110\times 25=₹2750 \end{gathered}$$