Question

(i) 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]
(ii) The radius and height of a solid right-circular cone are in the ratio of 5 : 12. If its volume is 314 cm³, find the total surface area. [Take π = 3.14.] [CBSE 2017]

Answer 1

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

(ii)
The ratio of radius and height of a solid right-circular cone is 5:12.
Let radius, r = 5x and height, h=12x.
Volume = 314 cm³.
$$\begin{gathered} \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^{2}h=314 \\ \Rightarrow \frac{1}{3}\times 3.14\times {\left(5x\right)}^2\times \left(12x\right)=314 \\ \Rightarrow x^3=\frac{314\times 3}{3.14\times 5\times 5\times 12} \\ \Rightarrow x^3=1\Rightarrow x=1 \end{gathered}$$
So, radius r = 5 cm and height h = 12 cm.
Using Pythagoras Theorem, slant height is given by $l=\sqrt{r^2+h^2}=\sqrt{25+144}=\sqrt{169}=13\mathrm{cm}$
Total Surface Area of Cone $=\mathrm{\pi }r\left(r+l\right)=3.14\times 5\times \left(5+13\right)=3.14\times 5\times 18=282.6{\mathrm{cm}}^2$.