Find the length of the string wound on a cylinder of height 48 cm and a base diameter of `5(1)/(11)` cm. The string makes exactly four complete turns round the cylinder while its two ends touch the cylinder’s top and bottom.

Options

A. 192 cm
B. 80 cm
C. 64 cm
D. Cannot be determined

Answer

To find the length of the string wound around a cylinder, we can follow these steps:

### Step 1: Determine the dimensions of the cylinder.
- The height of the cylinder (h) is given as 48 cm.
- The base diameter of the cylinder is given as \(5 \frac{1}{11}\) cm. We convert this to an improper fraction:
\[ 5 \frac{1}{11} = \frac{5 \times 11 + 1}{11} = \frac{56}{11} \text{ cm} \]

### Step 2: Calculate the radius of the cylinder.
- The radius (r) is half of the diameter:
\[ r = \frac{\text{Diameter}}{2} = \frac{56/11}{2} = \frac{28}{11} \text{ cm} \]

### Step 3: Calculate the circumference of the base of the cylinder.
- The circumference (C) can be calculated using the formula:
\[ C = 2 \pi r \]
Using \(\pi \approx \frac{22}{7}\):
\[ C = 2 \times \frac{22}{7} \times \frac{28}{11} \]
Simplifying this:
\[ C = \frac{44 \times 28}{77} = \frac{1232}{77} \approx 16 \text{ cm} \]

### Step 4: Determine the height per turn of the string.
- Since the string makes 4 complete turns around the cylinder, the height covered by each turn is:
\[ \text{Height per turn} = \frac{h}{4} = \frac{48}{4} = 12 \text{ cm} \]

### Step 5: Form a rectangle to visualize the string's path.
- When we unroll the cylinder, the string will form a diagonal in a rectangle where:
- One side (height) = 12 cm (height per turn)
- Other side (length) = circumference = 16 cm

### Step 6: Calculate the length of the string for one turn using the Pythagorean theorem.
- Using the Pythagorean theorem:
\[ \text{Length of string for one turn} = \sqrt{(\text{Height per turn})^2 + (\text{Circumference})^2} \]
\[ = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm} \]

### Step 7: Calculate the total length of the string for four turns.
- Since the string makes 4 turns:
\[ \text{Total length of string} = 4 \times 20 = 80 \text{ cm} \]

### Final Answer:
The length of the string wound around the cylinder is **80 cm**.

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