The cost of painting the surface of a solid cuboid of dimensions 40 cm × 36 cm × x cm at ₹ 20 per 100 cm² is ₹ 1,032. What is the value of x ?

The correct answer is 15.

Given parameters:
Dimensions of the solid cuboid: length $l=40\text{ cm}$, breadth $b=36\text{ cm}$, height $h=x\text{ cm}$.
Rate of painting = ₹ 20 per 100 cm².
Total cost of painting = ₹ 1,032.

Step 1: Find the total surface area of the cuboid.

The cost of painting per cm² is:

$\text{Rate per }{\text{cm}}^2=\frac{20}{100}=\text{₹ }0.2$

Since Total Cost = Total Surface Area × Rate per cm², we can write:

$\text{Total Surface Area}=\frac{\text{Total Cost}}{\text{Rate per }}$

$\text{Total Surface Area}=\frac{1032}{0.2}=\frac{1032\times 10}{2}=5160{\text{cm}}^2$

Step 2: Calculate the value of x using the total surface area formula.

The formula for the total surface area of a cuboid is:

$\text{Total Surface Area}=2(lb+bh+hl)$

Substitute the given values into the formula:

$2(40\times 36+36x+40x)=5160$

$2(1440+76x)=5160$

Divide both sides by 2:

$1440+76x=2580$

Subtract 1440 from both sides:

$76x=2580-1440$

$76x=1140$

Divide by 76:

$x=\frac{1140}{76}=15$

Thus, the value of x is 15 cm.