Question

The floor of a rectangular hall has a perimeter 250m. If the cost of pointing the four walls at the rate of Rs.10 per \[{{m}^{2}}\] is 15000, find the height of the hall (in meters).

Answer 1

Hint: Assume l, b and h as the length, breadth and height of the hall. Use the formula for the perimeter of a rectangle given as: - 2 (l + b) and equate it with 250 to form a relation. Now, divide the total cost of painting the four walls by the rate to find lateral surface area of the room, that is 2 (l + b) \[\times \] h. Finally, substitute the value of 2 (l + b) = 250 in the formula for L.S.A and get the value of h.

Complete step by step answer:
Let us assume the length, breadth and height of the room or hall are l, b and h respectively.
It is given that the perimeter of the floor of the rectangular hall is 250m. We know that, perimeter of a rectangle is given by the formula: - 2 (l + b). Therefore, we have,
\[\Rightarrow \] 2 (l + b) = 250 – (1)
Now, we have been provided with the information that the cost of painting the four walls of the hall at the rate of Rs. 10 per \[{{m}^{2}}\] is Rs. 15000. So, we have to paint four walls, that is the lateral surface of the hall. Here, we know that the total cost of painting is the product of the area to be painted and the rate of painting. So, we have,
Area to be painted = L.S.A of the hall
Rate = Rs. 10 per \[{{m}^{2}}\]
Total cost = Rs. 15000
Therefore, applying the formula for total cost, we get,
\[\Rightarrow \] Total cost = Rate \[\times \] L.S.A
\[\Rightarrow \] 15000 = 10 \[\times \] L.S.A
\[\Rightarrow \] L.S.A = 1500
We know that L.S.A of a cuboidal hall is given as: - 2 (l + b) \[\times \]h.
So, we have,
\[\Rightarrow 2\left( l+b \right)\times h~=1500\]
Substituting the value of 2 (l + b) = 250 from equation (1), we get,
\[\begin{align}
  & \Rightarrow 250\times h=1500 \\
 & \Rightarrow h=\dfrac{1500}{250} \\
 & \Rightarrow h=6m \\
\end{align}\]

Hence, the height of the room is 6m.

Note: One may note that we do not have to consider the total surface area (T.S.A) to find the height of the hall because the floor and ceiling of the hall is not painted. So, only four surfaces are to be painted, that is lateral surface area. Also note that here we do not have to try to find the values of ‘l’ and ‘b’ because we do not need that and there is not enough information regarding this.