Question

A conical tent was erected by the army at a base camp with height 3m and base diameter 8m. Find:
(i) The cost of canvas required for making a tent if the canvas cost Rs. 70 per 1 sq. m.
(ii) If every person requires 3.5 ${{m}^{3}}$ air, how many can be seated in that tent.

Answer 1

Hint: First, we have to identify which solid is used in this question. After knowing that cone is used we have to find radius which will be diameter by 2 and also will find slant height of cone using Pythagoras theorem ${{l}^{2}}={{r}^{2}}+{{h}^{2}}$ . Furthermore, in (i) we have to find how much cost will be there in using canvas to cover the cone shaped tent. So, in this curved surface area i.e. $\pi rl$ will be used. In (ii) we have to find how many person can be fitted in tent if one person requires 3.5 ${{m}^{3}}$ air, so in this volume of cone is to be find by using $\dfrac{1}{3}\pi {{r}^{2}}h$ .

Complete step-by-step answer:
Here, we are given that the height of the conical tent is 3m and the diameter of base is 8m. So, the radius of base will be $r=\dfrac{d}{2}=\dfrac{8}{2}=4m$ .

L is slant height of cone, which can be find using Pythagoras theorem i.e.,
${{l}^{2}}={{r}^{2}}+{{h}^{2}}$
${{l}^{2}}={{\left( 4 \right)}^{2}}+{{\left( 3 \right)}^{2}}$
${{l}^{2}}=16+9=25$
$\therefore l=\sqrt{25}=5m$ …………………………(i)
Now, taking case 1: we are given the cost of canvas used in the tent for 1 square meter. We have to find the overall cost of canvas used in a tent which can be given by using curved surface area.
Curved surface area (CSA) $=\pi rl$
Substituting the values in above equation, we get
$=\dfrac{22}{7}\times 4\times 5$
$=\dfrac{440}{7}{{m}^{2}}$
Therefore, CSA $=\dfrac{440}{7}{{m}^{2}}$ .
Now we are given a cost of 1 square meter Rs. 70. So, for find cost of 440 square meter we will use unitary method i.e.,
$\begin{align}
  & 1{{m}^{2}}=Rs.70 \\
 & \dfrac{440}{7}{{m}^{2}}=? \\
\end{align}$
$\Rightarrow \dfrac{440}{7}\times 70=Rs.4400$ …………………………….(ii)
Thus, Rs. 4400 is needed to cover the tent with canvas.
Now, taking case 2: Here, it is given that 3.5 cubic meter air is required for a single person in a tent. So, we have to find out how many people can stay in the tent. As, unit is cubic meters we can identify that here the volume of the tent is to be found.
$\therefore $ Volume $=\dfrac{1}{3}\pi {{r}^{2}}h$
$=\dfrac{1}{3}\times \dfrac{22}{7}\times {{\left( 4 \right)}^{2}}\times 3$
$=\dfrac{22}{7}\times 16$
$=\dfrac{352}{7}{{m}^{3}}$
Thus, the volume of the tent is $\dfrac{352}{7}{{m}^{3}}$ .So, now using a unitary method to find the number of persons.
$\begin{align}
  & 3.5{{m}^{3}}=1person \\
 & \dfrac{352}{7}{{m}^{3}}=? \\
\end{align}$
$\Rightarrow \dfrac{\dfrac{352}{7}}{3.5}=\dfrac{\dfrac{352}{7}}{\dfrac{7}{2}}$ (we know that 3.5 is equal to $\dfrac{7}{2}$ so, putting this for easy calculations)
$\Rightarrow \dfrac{352\times 2}{7\times 7}=14.36$ persons
Now, we know that persons can not be measured in decimal digits. So, considering that 14 persons can be in the tent.
Hence, Rs. 4400 is the cost of canvas and 14 persons can be seated in the tent.

Note: Be careful while selecting the curved surface area of cones because we can not put canvas on the bottom part, so formula for total surface area will not be used here, as it is having formula as $TSA=\pi r\left( r+l \right)$. If by mistake I use this, then the whole answer will go wrong. So, don’t make silly mistakes. Also, if you round off 14.36 as 15 instead of 14, then air will be less in the tent as per given data and it's better to be having extra air in the tent then extra 1 person. So, write answers accordingly.