Question

The height of a conical tent is 14 m and its floor area is 346.5 m². How much canvas, 1.1 wide, will be required for it?
(a) 490 m
(b) 525 m
(c) 665 m
(d) 860 m

Answer 1

(b) 525 m
Area of the floor of a conical tent$=\mathrm{\pi }{r}^2$
Therefore,

$$\begin{gathered} \mathrm{\pi }{r}^2=346.5 \\ \Rightarrow \frac{22}{7}\times r^2=346.5 \\ \Rightarrow r^2=\left(\frac{3465}{10}\times \frac{7}{22}\right) \\ \Rightarrow r^2=\frac{441}{4} \\ \Rightarrow r^2={\left(\frac{21}{2}\right)}^2 \\ \Rightarrow r=\frac{21}{2}\text{m} \end{gathered}$$

Height of the cone = 14 m

Slant height of the cone, $l=\sqrt{r^2+h^2}$

$$\begin{gathered} =\sqrt{{\left(\frac{21}{2}\right)}^2+{\left(14\right)}^2} \\ =\sqrt{\frac{441}{4}+196} \\ =\sqrt{\frac{1224}{4}} \\ =\frac{35}{4}\text{m} \end{gathered}$$

Area of the canvas = Curved surface area of the conical tent
$$\begin{gathered} =\mathrm{\pi }rl \\ =\left(\frac{22}{7}\times \frac{21}{2}\times \frac{35}{2}\right){\text{m}}^2 \\ =577.5{\text{m}}^2 \end{gathered}$$

Length of the canvas$=\frac{\text{Area of the canvas}}{\text{Width of the canvas}}$

$$\begin{gathered} =\frac{577.5}{1.1}\text{m} \\ =525\text{m} \end{gathered}$$