There is a wooden block in the form of a cube whose each side is $8$ meters long.

The maximum possible number of cylinders with a diameter of $1$ meter and a height of $4$ meters were cut from this block.

The cylinders are to be painted at the rate of $₹14$ per square meter.

What is the total amount (₹) needed to paint all the cylinders if we paint the entire surface of each cylinder?  (Take $\pi =\frac{22}{7}$ )

The correct answer is 25,344 (Option 3).

Step-by-step breakdown:

Step 1: Find the maximum number of cylinders that can be cut from the cube.
The wooden block is a cube with side length $8$ meters.
Each cylinder has a diameter of $1$ meter and a height of $4$ meters.
- Along the height of $8$ meters, we can stack $\frac{8}{4}=2$ layers of cylinders.
- Across the cross-sectional base of $8\times 8$ meters, the maximum number of $1$-meter diameter circles that fit is $8\times 8=64$ cylinders.
Therefore, the total maximum number of cylinders cut from the block is:

$\text{Total Cylinders}=64\times 2=128$

Step 2: Calculate the total surface area of one cylinder.
The radius of each cylinder is $r=\frac{1}{2}$ meter and height $h=4$ meters.
The formula for the total surface area of a cylinder is:

$\text{Surface Area}=2\pi r(h+r)$

Substitute $\pi =\frac{22}{7}$, $r=\frac{1}{2}$, and $h=4$:

$\text{Surface Area}=2\times \frac{22}{7}\times \frac{1}{2}\times \left(4+\frac{1}{2}\right)=\frac{22}{7}\times \frac{9}{2}=\frac{99}{7}{\text{m}}^2$

Step 3: Calculate the total cost to paint all 128 cylinders.
The painting rate is ₹ $14$ per square meter.

$\text{Total Cost}=128\times \frac{99}{7}\times 14$

Simplifying $\frac{14}{7}=2$:

$\text{Total Cost}=128\times 99\times 2=256\times 99=25,344$

Thus, the total amount needed to paint all the cylinders is ₹ 25,344.