The lateral surface area of a hollow cylinder is $$ 5000 \mathrm{~cm}^{2} $$. It is cut along its height and formed a rectangular sheet of width 50 cm . Find the perimeter of rectangular sheet.

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Given that the lateral surface area of a hollow cylinder is $$ 5000 \mathrm{~cm}^{2} $$ and it is cut along its height to form a rectangular sheet with a width of 50 cm, we need to find the perimeter of the rectangular sheet.

Let's denote the height of the cylinder as $$ h $$ and the radius of the cylinder as $$ r $$.

The lateral surface area of a hollow cylinder is given by the formula:
$$ 2\pi r h = 5000 $$

Since the cylinder is cut along its height, the width of the rectangular sheet is equal to the circumference of the base of the cylinder, which is $$ 2\pi r $$.

Given that the width of the rectangular sheet is 50 cm, we have:
$$ 2\pi r = 50 $$

Now, we can solve these equations to find the values of $$ h $$ and $$ r $$, and then calculate the perimeter of the rectangular sheet.

Let's solve the equations step by step.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}2\pi rh=5000\2\pi r=50\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}2\pi rh=5000\r=\frac{25}{\pi }\end{array}\right.$$
- step2: Substitute the value of $$r:$$
  $$2\pi \times \frac{25}{\pi }\times h=5000$$
- step3: Simplify:
  $$50h=5000$$
- step4: Divide both sides:
  $$\frac{50h}{50}=\frac{5000}{50}$$
- step5: Divide the numbers:
  $$h=100$$
- step6: Calculate:
  $$\left\{ \begin{array}{l}h=100\r=\frac{25}{\pi }\end{array}\right.$$
- step7: Check the solution:
  $$\left\{ \begin{array}{l}h=100\r=\frac{25}{\pi }\end{array}\right.$$
- step8: Rewrite:
  $$\left(h,r\right) = \left(100,\frac{25}{\pi }\right)$$
The solution to the system of equations is:
$$ h = 100 $$
$$ r = \frac{25}{\pi} $$

Now, we can calculate the perimeter of the rectangular sheet using the formula:
$$ \text{Perimeter} = 2 \times (\text{width} + \text{height}) $$

Substitute the values of width and height into the formula to find the perimeter.
Calculate the value by following steps:
- step0: Calculate:
  $$2\left(50+100\right)$$
- step1: Add the numbers:
  $$2\times 150$$
- step2: Multiply the numbers:
  $$300$$
The perimeter of the rectangular sheet is 300 cm.
The perimeter of the rectangular sheet is 300 cm.

The lateral surface area of a hollow cylinder is \( 5000 \mathrm{~cm}^{2} \). It is cut along its height and formed a rectangular sheet of width 50 cm . Find the perimeter of rectangular sheet.

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