Question

A hall $36\,m$ long and $24\,m$ broad allowing $80\,{m^2}$ for doors and windows, the cost of preparing the wall at ₹ 8.40 per ${m^2}$ is ₹ $9408. $Find the height of the hall.

Answer 1

Hint: Every wall will be rectangular in shape. The given information about the hall will be the lengths of adjacent walls. The height of all the walls will be the same. Calculate the total area of the walls in terms of height, $h$ of the wall. Subtract the area left for doors and windows. And the compare it with the cost of constructing those walls to find the value of $h$

Complete Step by Step Solution:
The hall will be made by 4 walls. The length of the opposite wall will be the same. And the height of all the 4 walls will be the same.
Now, let the length of the hall be the length of one wall, say $l = 36m$
Let the length of adjacent wall be the breadth of the hall, say $b = 24m$
Now in preparing the wall, we need to prepare the four walls excluding the floor and roof of the hall.
Area of doors and windows is given to be ${A_d} = 80{m^2}$ . . . . (1)
Now, the area of the two adjacent walls will be equal to
$lh + bh$
$\therefore $ Area of all the walls of the hall will be $A = 2(lh + bh)$
$ \Rightarrow A = 2(l + b)h$
$ \Rightarrow A = 2(36 + 24)h$
 $ \Rightarrow A = 120h\,{m^2}$ . . . . (2)
From equations (1) and (2)
Area of wall excluding the area of the windows and doors is given as
$A - {A_d} = (120h - 80)\,{m^2}$ . . . . . (3)
Cost of preparing the walls at ₹ 8.40 per ${m^2}$ = ₹ $9408.$ . . . . . (4)
From equation (3) and (4), we can say that the cost of preparing the complete area of the walls is given by,
$ \Rightarrow \left( {120h - 80} \right) \times 8.40 = 9408$
$ \Rightarrow 1008h - 672 = 9408$
Rearranging it we can write
$1008h = 9408 + 672$
$ \Rightarrow 1008h = 10,080$
$\therefore h = \dfrac{{10,080}}{{1008}}$
\[ \Rightarrow h = 10\,m\]

Hence, the height of each wall will be $10\,m$

Note:
Do not get confused with the length and breadth of the hall to be the length and breadth of the wall. You need to keep in mind that the hall will have two walls of different dimensions and the two opposite walls of the same dimensions. So, the length of the hall will be the length of one wall and the breadth of the hall will be the length of another wall. Cost of construction will be the product of the total area of construction and the cost of construction per unit area.