Question

Note Use $\mathrm{\pi }=\frac{22}{7}$, unless stated otherwise.
A cloth having an area of 165 m² is shaped into the form of a conical tent of radius 5 m. (i) How many students can sit in the tent if a student, on an average, iccupies $\frac{5}{7}{\mathrm{m}}^2$ on the ground? (ii) Find the volume of the cone.

Answer 1

Radius of the conical tent, r = 5 m

Area of the base of the conical tent = $\mathrm{\pi }{r}^2=\frac{22}{7}\times 5^2=\frac{550}{7}{\mathrm{m}}^2$

Average area occupied by a student on the ground = $\frac{5}{7}$ m²

∴ Number of students who can sit in the tent

$$\begin{gathered} =\frac{\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{base}\mathrm{of}\mathrm{the}\mathrm{conical}\mathrm{tent}}{\mathrm{Average}\mathrm{area}\mathrm{occupied}\mathrm{by}\mathrm{a}\mathrm{student}\mathrm{on}\mathrm{the}\mathrm{ground}} \\ =\frac{{\displaystyle \frac{550}{7}}}{{\displaystyle \frac{5}{7}}} \\ =110 \end{gathered}$$
Thus, the number of students who can sit in the tent is 110.

Let the slant height of the tent be l m.

Curved surface area of the tent = 165 m²

$$\begin{gathered} \therefore \mathrm{\pi }rl=165 \\ \Rightarrow \frac{22}{7}\times 5\times l=165 \\ \Rightarrow l=\frac{165\times 7}{22\times 5} \\ \Rightarrow l=10.5\mathrm{m} \end{gathered}$$
Let the height of the tent be h m.

$h=\sqrt{l^2-r^2}=\sqrt{{\left(10.5\right)}^2-5^2}=\sqrt{110.25-25}=\sqrt{85.25}$ ≈ 9.23 m

∴Volume of the tent = $\frac{1}{3}\mathrm{\pi }{r}^{2}h=\frac{1}{3}\times \frac{22}{7}\times {\left(5\right)}^2\times 9.23$ ≈ 241.74 m³

Thus, the volume of the conical tent is approximately 241.74 m³.