The curved surface of a cylinder is 100 sq cm .A wire of diameter 5mm is wound round it so as to cover it completely .Find the length of the wire.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the given information
We know that the curved surface area (CSA) of the cylinder is 100 sq cm, and the diameter of the wire is 5 mm. First, we need to convert the diameter of the wire into centimeters for consistency in units.

**Hint:** Remember that 1 cm = 10 mm.

### Step 2: Convert the diameter of the wire
The diameter of the wire is given as 5 mm. To convert this to centimeters:
\[ \text{Diameter of wire} = \frac{5 \text{ mm}}{10} = 0.5 \text{ cm} \]

### Step 3: Calculate the radius of the wire
The radius of the wire can be calculated as:
\[ \text{Radius of wire} = \frac{\text{Diameter}}{2} = \frac{0.5 \text{ cm}}{2} = 0.25 \text{ cm} \]

**Hint:** The radius is half of the diameter.

### Step 4: Use the formula for the curved surface area of the cylinder
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
where \( r \) is the radius of the cylinder and \( h \) is the height of the cylinder.

### Step 5: Set up the equation
We know the CSA is 100 sq cm. Therefore:
\[ 2 \pi r h = 100 \]

### Step 6: Relate the height of the cylinder to the wire
The height of the cylinder can be expressed in terms of the number of turns of the wire \( n \):
\[ h = n \times \text{Diameter of wire} = n \times 0.5 \text{ cm} \]

### Step 7: Substitute height in the CSA equation
Substituting \( h \) in the CSA equation:
\[ 2 \pi r (n \times 0.5) = 100 \]
This simplifies to:
\[ \pi r n = \frac{100}{1} = 200 \]

### Step 8: Calculate the circumference of the cylinder
The circumference of the cylinder is given by:
\[ C = 2 \pi r \]

### Step 9: Find the length of the wire
The length of the wire \( L \) can be expressed as:
\[ L = C \times n = (2 \pi r) \times n \]

### Step 10: Substitute \( n \) from the CSA equation
From the equation \( \pi r n = 200 \), we can express \( n \):
\[ n = \frac{200}{\pi r} \]

Now substituting this into the length of the wire:
\[ L = (2 \pi r) \times \left(\frac{200}{\pi r}\right) = 2 \times 200 = 400 \text{ cm} \]

### Final Answer
Thus, the length of the wire is:
\[ \text{Length of the wire} = 400 \text{ cm} \]