10.1 The outboard motor (pictured below) is used to propel boats through water and has a 4-stroke engine - NSC Technical Mathematics - Question 10 - 2022 - Paper 2

The relationship between linear velocity ($V$) and angular velocity ($\omega$) is given by:

$$V = r \omega$$

Thus:

$$\omega = \frac{V}{r} = \frac{8.33}{180} \approx 0.0463 \, \text{rad/s}$$

To convert degrees to radians, we use the conversion:

$$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$$

Therefore:

$$210° = 210 \times \frac{\pi}{180} = \frac{7\pi}{6} \, \text{rads}$$

The length of an arc is calculated using the formula:

$$s = r\theta$$

For major arc BC:

$$s = 5 \times \frac{7\pi}{6} = \frac{35\pi}{6} \approx 18.33 \, \text{cm}$$

Given the area of the shaded sector:

$$A = \frac{1}{2}r^2\theta$$

Substituting for area:

$$54 = \frac{1}{2}(9)^2\theta$$

This leads to:

$$\theta = \frac{54 \times 2}{81} = \frac{108}{81} = \frac{4\pi}{3} \, \text{rads}$$

Using the half-chord method:

$$FDE = \sqrt{(r^2 - d^2)}$$

where $r$ is the radius and $d$ is half of chord length. The values give:

$$EF = 2 \sqrt{9^2 - 7^2} \approx 17.55 \, ext{cm}$$

To find the length of the rubber belt not in contact, subtract the length of arcs in contact from the total length:

Total length = 140 cm Length in contact = Length of arc BC (18.33 cm) + Length of arc JK (4.19 cm) + Length of arc EF (37.70 cm)

Thus, the non-contact length = 140 - (18.33 + 4.19 + 37.70) = 140 - 60.22 = 79.78 cm.