Calculate Cylinder Diameter: Sheet Rolled into Solid Cylinder

This solution determines the diameter of a solid cylinder formed by rolling a sheet, using the principle of volume conservation.

Given Sheet Dimensions

• Length: $10 \text{ km} = 10,000 \text{ m}$
• Width: $1 \text{ m}$
• Thickness: $0.1 \text{ mm} = 10^{-4} \text{ m}$

Calculate Sheet Volume

First, find the volume ($V$) of the sheet material. The volume of the resulting solid cylinder must be equal to this.

$ V = \text{Length} \times \text{Width} \times \text{Thickness} $

$ V = (10,000 \text{ m}) \times (1 \text{ m}) \times (10^{-4} \text{ m}) $

$ V = 1 \text{ m}^3 $

Cylinder Volume and Diameter Relationship

The volume of a cylinder is given by the formula $ V = \pi r^2 h $. Using the diameter $d$ (where $r = d/2$), the formula is:

$ V = \pi \left(\frac{d}{2}\right)^2 h = \frac{\pi d^2 h}{4} $

For the cylinder formed, we assume the sheet's width ($W$) becomes the cylinder's height ($h$):

$ h = W = 1 \text{ m} $

Solving for Cylinder Diameter

Equate the calculated sheet volume ($1 \text{ m}^3$) to the cylinder volume formula and solve for the diameter $d$:

$ V_{\text{sheet}} = V_{\text{cylinder}} $

$ 1 \text{ m}^3 = \frac{\pi d^2 h}{4} $

Substitute the value $ h = 1 \text{ m} $ into the equation:

$ 1 = \frac{\pi d^2 (1)}{4} $

Rearrange to solve for $d^2$:

$ \pi d^2 = 4 $

$ d^2 = \frac{4}{\pi} $

Take the square root of both sides to find the diameter:

$ d = \sqrt{\frac{4}{\pi}} = \frac{2}{\sqrt{\pi}} $

$ d = \frac{2}{\pi^{1/2}} \text{ m} $