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 side length of the cube-shaped auditorium
Let the side length of the auditorium be $s$ meters.
The total area of the four interior vertical walls of a cube-shaped auditorium is given by:

$\text{Area of 4 walls}=4s^2$

Given that the rate of painting is 15 Rs./m² and the total cost is Rs. 8640:

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

$8640=4s^2\times 15$

$8640=60s^2$

$s^2=\frac{8640}{60}=144$

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

Step 2: Determine dimensions of the rectangular storage hall

Length ($L$) = $s+2=12+2=14\text{m}$

Width ($W$) = $s-2=12-2=10\text{m}$

Height ($H$) = $s=12\text{m}$

Step 3: Calculate total area of the four interior vertical walls of the storage hall
The area of four vertical walls of a rectangular hall (cuboid) is calculated as:

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

Substitute the dimensions into the formula:

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

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

Step 4: Calculate total expense for painting the storage hall
At the rate of 15 Rs./m²:

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

Shortcut Method:
Algebraically, the area of the 4 vertical walls of the rectangular hall is:

$2\times \left(\right(s+2)+(s-2\left)\right)\times s=2\times \left(2s\right)\times s=4s^2$

Since the vertical wall area remains $4s^2$, the total painting cost remains unchanged at Rs. 8640.