(a) Two cyclists X and Y leave town Q at the same time. Cyclist X travels at the rate of 5 km/h on a bearing of 049° and cyclist Y travels at the rate of 9 km/h on a bearing of 319°.
(a) Illustrate the information on a diagram.
(b) After travelling for two hours, calculate. correct to the nearest whole number, the:
(i) distance between cyclist X and Y;
(ii) bearing of cyclist X from Y.
(c) Find the average speed at which cyclist X will get to Y in 4 hours.
(a) The diagram can be drawn by placing town Q at the origin and drawing the bearings of 049° and 319°. Then, draw lines from town Q at those bearings to represent the paths of cyclists X and Y.
(b)
- (i) To find the distance between cyclist X and Y, we need to use the cosine rule. Let a be the distance that cyclist X travels and b be the distance that cyclist Y travels. Then, we have:
- a² = (2 × 5)² = 100
- b² = (2 × 9)² = 324
- c² = a² + b² - 2ab cos(128°) ? 302.73
- c ? 17.4 km (nearest whole number)
- (ii) To find the bearing of cyclist X from Y, we need to use the sine rule. Let C be the angle between cyclist X's path and the line connecting X and Y, and let D be the angle between cyclist Y's path and the same line. Then, we have:
- sin(C) / a = sin(128°) / c
- sin(D) / b = sin(33°) / c
Solving for C and D, we get:
Since cyclist X is to the right of cyclist Y, the bearing of cyclist X from Y is:
- 360° - 319° + 97.2° ? 138.2°
(c) To find the average speed at which cyclist X will get to Y in 4 hours, we need to find the total distance between them and divide by the time taken. Using the distance formula from part (b)(i), we have:
d = ?302.73 ? 17.4 km
time taken = 4 hours
Therefore, the average speed at which cyclist X will get to Y in 4 hours is:
17.4 km / 4 hours = 4.35 km/h
(a) The diagram can be drawn by placing town Q at the origin and drawing the bearings of 049° and 319°. Then, draw lines from town Q at those bearings to represent the paths of cyclists X and Y.
(b)
- (i) To find the distance between cyclist X and Y, we need to use the cosine rule. Let a be the distance that cyclist X travels and b be the distance that cyclist Y travels. Then, we have:
- a² = (2 × 5)² = 100
- b² = (2 × 9)² = 324
- c² = a² + b² - 2ab cos(128°) ? 302.73
- c ? 17.4 km (nearest whole number)
- (ii) To find the bearing of cyclist X from Y, we need to use the sine rule. Let C be the angle between cyclist X's path and the line connecting X and Y, and let D be the angle between cyclist Y's path and the same line. Then, we have:
- sin(C) / a = sin(128°) / c
- sin(D) / b = sin(33°) / c
Solving for C and D, we get:
Since cyclist X is to the right of cyclist Y, the bearing of cyclist X from Y is:
- 360° - 319° + 97.2° ? 138.2°
(c) To find the average speed at which cyclist X will get to Y in 4 hours, we need to find the total distance between them and divide by the time taken. Using the distance formula from part (b)(i), we have:
d = ?302.73 ? 17.4 km
time taken = 4 hours
Therefore, the average speed at which cyclist X will get to Y in 4 hours is:
17.4 km / 4 hours = 4.35 km/h