A cylinder 84 cm high has a circumference of 16 cm. A string makes exactly 7 complete turns round the cylinder while its two ends touch the cylinder's top and bottom. How long is the string in cm?

Options

A. 160
B. 190
C. 140
D. 180

Answer

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

### Step 1: Understand the problem
We have a cylinder with a height of 84 cm and a circumference of 16 cm. A string wraps around the cylinder making 7 complete turns from the top to the bottom.

### Step 2: Calculate the height of one turn
To find the height covered by the string in one complete turn, we divide the total height of the cylinder by the number of turns:
\[ \text{Height of one turn} = \frac{\text{Total height}}{\text{Number of turns}} = \frac{84 \text{ cm}}{7} = 12 \text{ cm} \]

### Step 3: Use the helical length formula
The length of the string can be calculated using the helical length formula:
\[ L = \sqrt{h^2 + C^2} \]
where \( h \) is the height of one turn and \( C \) is the circumference of the cylinder.

### Step 4: Substitute the values into the formula
We already calculated \( h = 12 \text{ cm} \) and \( C = 16 \text{ cm} \). Now we substitute these values into the formula:
\[ L = \sqrt{(12 \text{ cm})^2 + (16 \text{ cm})^2} \]
\[ L = \sqrt{144 \text{ cm}^2 + 256 \text{ cm}^2} \]
\[ L = \sqrt{400 \text{ cm}^2} \]
\[ L = 20 \text{ cm} \]

### Step 5: Calculate the total length of the string
Since the string makes 7 complete turns, the total length of the string is:
\[ \text{Total length} = \text{Length of one turn} \times \text{Number of turns} = 20 \text{ cm} \times 7 = 140 \text{ cm} \]

### Final Answer
The length of the string is **140 cm**.

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