A cylinder and its net are shown below. a) Work out the width, w, of the rectangle. b) Calculate the area of the curved surface of the cylinder. Give your answers to the nearest integer and remember to give the correct units.

Part (a)

Step 1: The rectangle in the net is the lateral surface of the cylinder.
Its width equals the circumference of the base, and its length equals the height of the cylinder ($10$ cm).

Formula for circumference of the base:
$C = 2\pi r$
where $r = 7$ cm (from the diagram).

Step 2: Substitute the value of $r$:
$C = 2\pi \times 7 = 14\pi \text{ cm}$

Step 3: Approximate using $\pi \approx 3.1416$:
$14 \times 3.1416 = 43.9824 \approx 44 \text{ cm (to the nearest cm)}$

The width of the rectangle is $\boxed{44 cm}$.

Part (b)

Curved surface area (lateral surface area):

Step 1: Formula:
$A_{curved} = 2\pi r h$
where $r = 7$ cm, $h = 10$ cm.

Step 2: Substitute the values:
$A_{curved} = 2\pi \times 7 \times 10 = 140\pi cm^2$

Step 3: Approximate using $\pi \approx 3.1416$:
$140 \times 3.1416 = 439.824 \approx 440 cm^2 \text{ (to the nearest cm)}$

Total surface area:

Step 1: Formula:
$A_{total} = 2\pi r h + 2\pi r^2$

Step 2: First, curved surface area $= 140\pi$ cm$^2$ (from above).
Area of two bases:
$2\pi r^2 = 2\pi \times 7^2 = 2\pi \times 49 = 98\pi cm^2$

Step 3: Total:
$A_{total} = 140\pi + 98\pi = 238\pi cm^2$

Step 4: Approximate:
$98 \times 3.1416 = 307.8768$
$140 \times 3.1416 + 98 \times 3.1416 = 439.824 + 307.8768 = 747.7008 \approx 748 cm^2 \text{ (to the nearest cm)}$

Curved surface area: $\boxed{440 cm^2}$
Total surface area: $\boxed{748 cm^2}$