Geometry Question: A student winds a strip of paper eight times round a cylindrical pencil of diameter 7mm. Use the value \( \frac{22}{7}\) for \(\pi\) to find the length of the paper.

A student winds a strip of paper eight times round a cylindrical pencil of diameter 7mm. Use the value \( \frac{22}{7}\) for \(\pi\) to find the length of the paper.

 Solution

Step 1: Understand the Geometry

To solve the problem, we need to understand the geometry of the cylinder. A cylindrical pencil has:

• Diameter (d): 7 mm
• Radius (r): Half of the diameter, so \(r=\frac{7}{2}=3.5mm\)
• Number of Turns: The paper is wound around the pencil 8 times.

Each time the paper is wrapped around the pencil, it traces a circular path along the surface of the pencil. The length of the paper wrapped in one complete turn corresponds to the circumference of the pencil.

Step 2: Calculate the Circumference of the Pencil

The circumference of a circle is given by the formula:

$$C = 2 \pi r$$where:

• $C$ is the circumference,
• $r$ is the radius of the pencil,
• and $\pi$ is approximately $\frac{22}{7}$ as per the problem's instructions.

Now, we can substitute the known values:

$$C = 2 \times \frac{22}{7} \times 3.$$First, multiply the values inside the parentheses:$$C = 2 \times \frac{22}{7} \times 3.5 = 2 \times \frac{77}{7} = 2 \times 11 = 22 \, \text{mm$$So, the circumference of the pencil is 22 mm.

Step 3: Account for the Number of Turns

Since the paper is wrapped 8 times around the pencil, the total length of the paper will be the circumference multiplied by the number of turns:

$$\text{Total Length} = 8 \times 22 \, \text{mm} = 176 \, \text{mm}$$

Step 4: Final Answer

The total length of the paper used is 176 mm.