The curved surface of a cylinder is `1000" cm"^(2)`. A wire of diameter 5 mm is wound around it, so as to cover it completely. What is the length of the wire used ?

Options

A. 22 m
B. 20 m
C. 18 m
D. None of these

Answer

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

### Step 1: Understand the given information
We are given the curved surface area (CSA) of a cylinder, which is \(1000 \, \text{cm}^2\), and the diameter of the wire, which is \(5 \, \text{mm}\). We need to find the length of the wire used to completely cover the cylinder.

### Step 2: Convert the diameter of the wire to centimeters
The diameter of the wire is given in millimeters. We need to convert it to centimeters for consistency in units:
\[ \text{Diameter of wire} = 5 \, \text{mm} = \frac{5}{10} \, \text{cm} = 0.5 \, \text{cm} \]

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

### 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: Rearranging the formula to find \(h\)
From the CSA, we can express \(h\) as:
\[ h = \frac{\text{CSA}}{2 \pi r} \]
Substituting the given CSA:
\[ h = \frac{1000}{2 \pi r} \]

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

### Step 7: Set the two expressions for height equal to each other
Equating the two expressions for height:
\[ n \times 0.5 = \frac{1000}{2 \pi r} \]
From this, we can express \(n\):
\[ n = \frac{1000}{2 \pi r \times 0.5} = \frac{1000}{\pi r} \]

### Step 8: Calculate the length of the wire
The length of the wire \(L\) is given by the formula:
\[ L = \text{Circumference of the cylinder} \times n \]
The circumference of the cylinder is:
\[ \text{Circumference} = 2 \pi r \]
Thus, substituting for \(L\):
\[ L = (2 \pi r) \times n \]
Substituting \(n\) from the previous step:
\[ L = (2 \pi r) \times \left(\frac{1000}{\pi r}\right) \]
The \(\pi r\) cancels out:
\[ L = 2 \times 1000 = 2000 \, \text{cm} \]

### Step 9: Convert the length from centimeters to meters
To convert centimeters to meters:
\[ L = \frac{2000}{100} = 20 \, \text{meters} \]

### Final Answer
The length of the wire used is \(20 \, \text{meters}\).

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