The total cost to paint the interior walls of a cube-shaped auditorium is Rs.8640 when charged at a rate of 15 Rs./m². A rectangular storage hall is constructed such that its length is 2 m longer and its width is 2 m shorter than the side length of the auditorium, while its height is equal to the side length of the auditorium. Calculate the total expense to paint the four interior vertical walls of the rectangular storage hall at the exact same rate per square meter.

The correct answer is Rs.8640.

Step 1: Find the wall area and side length of the cube-shaped auditorium
A cube-shaped auditorium has 4 interior vertical walls of equal square area. Let the side length of the auditorium be s meters.
First, we find the total area of the 4 vertical walls of the auditorium using the given total painting cost and rate per square meter:

$\text{Area of 4 walls}=\frac{\text{Total Cost}}{\text{Rate per m}{2}^{\mathrm{}}}$

$\text{Area of 4 walls}=\frac{8640}{15}=576{\text{m}}^2$

Since the area of 4 vertical square walls of side length s is equal to 4s², we set up the equation:

$4s^2=576$

$s^2=\frac{576}{4}=144$

$s=\sqrt{144}=12\text{m}$

Thus, the side length of the auditorium is 12 m.

Step 2: Determine the dimensions of the rectangular storage hall
Using the side length s = 12 m, we calculate the dimensions of the rectangular storage hall:
• Length (L) = s + 2 = 12 + 2 = 14 m
• Width (W) = s - 2 = 12 - 2 = 10 m
• Height (H) = s = 12 m

Step 3: Calculate the area of the 4 vertical walls of the storage hall
The formula for the total area of the four vertical walls of a rectangular room (lateral surface area) is given by:

$\text{Area of 4 walls}=2\times (L+W)\times H$

Substituting the calculated dimensions into the formula:

$\text{Area of 4 walls}=2\times (14+10)\times 12$

$\text{Area of 4 walls}=2\times 24\times 12=576{\text{m}}^2$

Note: Notice that algebraically, the area of the four walls of the rectangular hall is $2\times \left(\right(s+2)+(s-2\left)\right)\times s=4s^2$, which is identical to the total vertical wall area of the original cube-shaped auditorium.

Step 4: Calculate the total cost to paint the storage hall
At the exact same rate of 15 Rs./m², the total expense is:

$\text{Total Expense}=\text{Area}\times \text{Rate}$

$\text{Total Expense}=576\times 15=\text{Rs.8640}$

Therefore, the total expense to paint the four interior vertical walls of the rectangular storage hall is Rs.8640.