Let's break down the problem step by step based on the image provided.

Part (a): Work out the width $w$ of the rectangle.

The width $w$ of the rectangle corresponds to the circumference of the circular top of the cylinder.

The circumference $C$ of a circle is given by the formula: $C = 2 \pi r$ where $r$ is the radius of the cylinder.

From the diagram:

• The diameter of the circle is $7$ cm, so the radius $r = \frac{7}{2} = 3.5$ cm.

Now, substituting into the formula for the circumference: $w = 2 \pi \times 3.5$ $w \approx 2 \times 3.1416 \times 3.5 \approx 21.99 \, \text{cm}$

To the nearest integer: $w \approx 22 \, \text{cm}$

Part (b): Calculate the area of the curved surface of the cylinder.

The curved surface area $A$ of a cylinder is given by the formula: $A = 2 \pi r h$ where:

• $r$ is the radius of the cylinder ($r = 3.5$ cm),
• $h$ is the height of the cylinder ($h = 11$ cm).

Substituting the values: $A = 2 \pi \times 3.5 \times 11$ $A \approx 2 \times 3.1416 \times 3.5 \times 11 \approx 241.91 \, \text{cm}^2$

To the nearest integer: $A \approx 242 \, \text{cm}^2$

Final Answers:

• The width $w$ of the rectangle is approximately 22 cm.
• The curved surface area of the cylinder is approximately 242 cm².

Would you like further clarification or a detailed explanation of any steps?

Here are some related questions:

1. How is the formula for the circumference of a circle derived?
2. What is the relationship between the diameter and the radius of a circle?
3. How would the surface area change if the height of the cylinder doubled?
4. What is the formula for the total surface area of a cylinder, including its top and bottom?
5. How would the area change if the radius were halved?

Tip: Always double-check your unit conversions, especially when dealing with different measurements like diameter and radius.