Question

A cloth having an area of $165{\mathrm{m}}^2$ is shaped into a form of conical tent of radius 5 m.

(i) How many students can sit in the tent if a student on an average, occupies $\frac{5}{7}{\mathrm{m}}^2$ on the ground?

(ii) Find the capacity of tent.

Answer 1

Step 1: Finding the number of students

Total area of seating would be equal to the base area of the cone.

Since a cone has circular base, therefore

Base area of the cone,

$A={\mathrm{\pi r}}^2$, where ‘r’ is the radius of the cone.

Therefore,

$\mathrm{A}=\frac{22}{7}\times 5^2[\mathrm{Here},\mathrm{r}=5\mathrm{m}]$

Thus,

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{student}=\frac{\mathrm{Total}\mathrm{base}\mathrm{area}}{\mathrm{Area}\mathrm{occupied}\mathrm{by}\mathrm{one}\mathrm{student}} \\ =\frac{{\displaystyle \frac{22}{7}\times 5^2}}{{\displaystyle \frac{5}{7}}} \\ =110\mathrm{Students} \end{gathered}$$

Hence, the number of students are 110.

Step 2: Finding the capacity of tent(V)

We have:

Curved surface area of cone = area of cloth

Therefore,

$$\begin{gathered} \mathrm{\pi rl}=165 \\ \Rightarrow \frac{22}{7}\times 5\times \mathrm{l}=165[\mathrm{Here},\mathrm{l}\mathrm{is}\mathrm{the}\mathrm{slant}\mathrm{height},\mathrm{\pi }=\frac{22}{7}] \\ \Rightarrow \mathrm{l}=\frac{165\times 7}{110} \\ \Rightarrow \mathrm{l}=10.5\mathrm{m} \end{gathered}$$

And,

Height ‘h’ of the cone can be calculated as:

$$\begin{gathered} \mathrm{l}=\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2} \\ \Rightarrow \mathrm{h}=\sqrt{{\mathrm{l}}^2-{\mathrm{r}}^2} \\ \Rightarrow \mathrm{h}=\sqrt{{(10.5)}^2-5^2} \\ \Rightarrow \mathrm{h}=\sqrt{85.25}=9.23\mathrm{m} \end{gathered}$$

Now, the capacity of tent can be calculated as:

$V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

Substituting the values we get,

$$\begin{gathered} \mathrm{V}=\frac{1}{3}\times \frac{22}{7}\times 5^2\times 9.23 \\ =241.73{\mathrm{m}}^2 \end{gathered}$$

Therefore, the capacity of tent is $241.73{\mathrm{m}}^2$.