Question

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

(I) How many students can sit in the tent if a student occupies $\mathbf{5}\mathbf{/}\mathbf{7}\mathbf{}{\mathbf{m}}^{\mathbf{2}}$.

(ii) Find the volume of air for each student.

Answer 1

Step 1: Given information:

The base radius of the conical tent $\left(r\right)=5\mathrm{m}$

The area occupied by a student$=5/7{\mathrm{m}}^2$

Step 2: Calculate the number of students.

First, we have to find the base area of the conical tent.

The base area$\left(A\right)$ is computed as:

$$\begin{gathered} \Rightarrow A=\mathrm{\pi }{r}^2 \\ \Rightarrow A=\frac{22}{7}\times 5\times 5 \\ \Rightarrow A=\frac{550}{7}{\mathrm{m}}^2 \end{gathered}$$

Since the area that is occupied by a student$=5/7{\mathrm{m}}^2$,

$\therefore$The number of students that sit in the tent$=\frac{550}{7}\times \frac{7}{5}=110$

Step 3: Calculate the height of the conical tent.

Since a cloth has an area of $165{\mathrm{m}}^2$, which is shaped into a conical tent of the radius $5\mathrm{m}$.

$\therefore$ The area of the conical tent$=165$

$$\begin{gathered} \therefore \mathrm{\pi }rl=165 \\ \Rightarrow \frac{22}{7}\times 5\times l=165 \\ \Rightarrow l=\frac{165\times 7}{22\times 5} \\ \Rightarrow l=10.5\mathrm{m} \end{gathered}$$

Thus, the slant height $\left(l\right)$ of the conical tent is $10.5\mathrm{m}$.

$\therefore$ The height$\left(h\right)$ of the conical tent is computed as:

$$\begin{gathered} \Rightarrow h^2=l^2-r^2 \\ \Rightarrow h^2={(10.5)}^2-{\left(5\right)}^2\Rightarrow h^2=110.25-25 \\ \Rightarrow h^2=85.25 \\ \Rightarrow h=\sqrt{85.25}=9.23\mathrm{m} \end{gathered}$$

Step 4: Calculate the volume of air for each student.

$$\begin{gathered} \therefore V=\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ \Rightarrow V=\frac{1}{3}\times \frac{22}{7}\times {\left(5\right)}^2\times 9.23 \\ \Rightarrow V=241.74{\mathrm{m}}^3 \end{gathered}$$

Thus, the volume of air for each student$=\frac{241.74}{110}=2.198{\mathrm{m}}^3$.

Hence,

(I) The number of students that sit in the tent is $110$.

(ii) The volume of air for each student is $2.198{\mathrm{m}}^3$.