Question

The length of a hall is 20m and width 16m. The sum of the areas of the floor and the flat roof is equal to the sum of the area of the four walls. Find the height of the hall.

Answer 1

Hint:Area of a rectangular surface is given as \[A = l \times b\]
In this question since the given dimensions length and the width are different so use the formula of area of rectangle and find the area of floor, roof and all the walls.

As the sum of the areas of the floor and the flat roof is equal to the sum of the area of the four walls by substituting the values find the height of the hall.

Complete step by step solution:
Given The length of the hall \[l = 20m\]
Width of the hall\[b = 16m\]

Let the height of the hall be\[h\]
Also let the area of the floor of hall be\[x\]

Now since the length and the width of the floor of the hall is not same so the we can say the hall is a rectangle hall, hence the area of the floor of the room will be
\[x = l \times b = 20 \times 16 = 320{m^2}\]

Now we will find the sum of area of the four walls of the hall and since the hall is a rectangle hall so the areas of the walls will be
\[
area = \left( {l \times h + b \times h + l \times h + b \times h} \right) \\
= 2l \times h + 2b \times h \\
= 2h\left( {l + b} \right) \\
\]

Now it is said that the sum of the areas of the floor and the flat roof where the area of the floor is equal to the area of the roof is equal to the sum of the area of the four walls, hence we can write
\[
2x = 2h\left( {l + b} \right) \\

x = h\left( {l + b} \right) \\
\]
Now by substituting the values in the equation, we get
\[
x = h\left( {l + b} \right) \\
320 = h\left( {20 + 16} \right) \\
320 = 36h \\
h = \dfrac{{320}}{{36}} \\
= 8.89m \\
\]

Hence the height of the hall\[h = 8.89m\]

Note:This technique to find the area of each wall and the floor and the roof of a room is used in civil construction sites to find the estimate of the construction. Engineers generally find the areas of the room by this technique and the cost of the work.
When length, breadth and the height of a room are multiplied together we get the volume of the room.
\[V = l \times b \times h\]