1. **Problem (c):** Determine the number of complete revolutions a motorcycle wheel will make in travelling 2 km, given the wheel’s diameter is 85.1 cm. 2. **Formula:** The distance travelled by the wheel equals the number of revolutions multiplied by the circumference of the wheel. $$\text{Distance} = \text{Number of revolutions} \times \text{Circumference}$$ 3. **Calculate the circumference:** The circumference $C$ of a circle is given by: $$C = \pi \times d$$ where $d$ is the diameter. Given $d = 85.1$ cm, convert to meters: $$85.1\,\text{cm} = 0.851\,\text{m}$$ So, $$C = \pi \times 0.851 = 2.673\,\text{m}$$ (approx.) 4. **Convert distance travelled to meters:** $$2\,\text{km} = 2000\,\text{m}$$ 5. **Calculate number of revolutions:** $$\text{Number of revolutions} = \frac{\text{Distance}}{\text{Circumference}} = \frac{2000}{2.673} \approx 748.3$$ Since only complete revolutions count, the number of complete revolutions is: $$748$$ --- 6. **Problem (d):** Given current values at equal time intervals, plot current against time and estimate the area under the curve over 30 ms using the mid-ordinate rule, then determine the mean value of the current. 7. **Data:** | Time (ms) | 0 | 5 | 10 | 15 | 20 | 25 | 30 | |-----------|---|---|----|----|----|----|----| | Current (A) | 0 | 0.9 | 2.6 | 4.9 | 5.8 | 3.5 | 0 | 8. **Mid-ordinate rule:** The mid-ordinate rule estimates area under a curve by taking the function values at midpoints of intervals. Intervals are 5 ms each, so midpoints are at 2.5, 7.5, 12.5, 17.5, 22.5, 27.5 ms. We approximate the current at midpoints by averaging the current values at the endpoints of each interval: - Interval 0-5 ms: midpoint current = $(0 + 0.9)/2 = 0.45$ A - Interval 5-10 ms: midpoint current = $(0.9 + 2.6)/2 = 1.75$ A - Interval 10-15 ms: midpoint current = $(2.6 + 4.9)/2 = 3.75$ A - Interval 15-20 ms: midpoint current = $(4.9 + 5.8)/2 = 5.35$ A - Interval 20-25 ms: midpoint current = $(5.8 + 3.5)/2 = 4.65$ A - Interval 25-30 ms: midpoint current = $(3.5 + 0)/2 = 1.75$ A 9. **Calculate area under the curve:** Area $\approx \sum (\text{midpoint current} \times \text{interval width})$ Interval width = 5 ms = 0.005 s (if needed for units, but here ms is consistent) Sum of midpoint currents: $$0.45 + 1.75 + 3.75 + 5.35 + 4.65 + 1.75 = 17.7$$ Area: $$17.7 \times 5 = 88.5\,\text{A·ms}$$ 10. **Mean value of current:** Mean current $= \frac{\text{Area under curve}}{\text{Total time}}$ Total time = 30 ms $$\text{Mean current} = \frac{88.5}{30} = 2.95\,\text{A}$$ --- **Final answers:** - (c) Number of complete revolutions = $748$ - (d) Area under curve (approx.) = $88.5$ A·ms - Mean current = $2.95$ A