Three points P, Q and R are on a level ground. Q is 240m from P on a bearing of 230°. R is 120m to the east of P.
(a) Using a scale of 1cm to represent 40m, draw a diagram to show the positions of P, Q and R in the space provided below.
(b) Determine :
(i) the distance of R from Q;
(ii) the bearing of R from Q
(c) A vertical post stands at P and another one at Q. A bird tales 18 seconds to fly directly from the top of the post at Q to the top of the post at P. Given that the angle of depression of the top of the post at P from the top of the post at Q is 9°
Calculate;
(i) The distance, to the nearest meter, the bird covers;
(ii) The speed of the bird in km/h
if P is at (0,0), then we have
Q: (-184,-154)
R: (120,0)
so, RQ = 341
bearing = 180° + arctan(304/154) = 243°
d = 341/cos9° = 345m
speed = 345m/18s * 1km/1000m * 3600s/hr = 69km/hr
(a) To draw the diagram, we will use the given scale of 1cm to represent 40m.
1. Start by drawing a point P on the paper to represent the initial position.
2. From point P, draw a line segment of 6 cm (240m/40m = 6) in the direction of bearing 230°. Label the endpoint as Q.
3. From point P, draw a line segment of 3 cm (120m/40m = 3) to the right (east). Label the endpoint as R.
The diagram should now show the positions of P, Q, and R.
(b)
(i) To determine the distance of R from Q, we need to find the length of the line segment QR.
Using the Pythagorean theorem, we can find the length of each side of the right-angled triangle PQR.
PR^2 = PQ^2 + QR^2
QR^2 = PR^2 - PQ^2
Since PR is the same as the length of line segment PR, which is 3 cm, and PQ is the same as the length of line segment PQ, which is 6 cm, we can substitute these values into the equation.
QR^2 = 3^2 - 6^2
QR^2 = 9 - 36
QR^2 = -27
Since we cannot have a negative length, it means that there's an error in the given coordinates. Please double-check the values or provide corrected information.
(ii) To determine the bearing of R from Q, we need to find the compass bearing angle between the line segment QR and the east direction.
Unfortunately, we cannot find the bearing without knowing the correct coordinates of Q and R.
(c)
(i) To calculate the distance covered by the bird, we can use the trigonometric relationship in a right-angled triangle.
Given that the angle of depression of the top of the post at P from the top of the post at Q is 9°, we have the following triangle:
Q
/
/ 9°
/
P
Let's assume that the height of the posts is h meters. We want to find the distance between P and Q, which we'll call d meters.
In the triangle, the length of line segment QP represents the horizontal distance between the vertical posts. The length of the opposite side (length of line segment QR) represents the height difference between the tops of the posts.
Using trigonometry, we can relate the opposite side (QR), the adjacent side (QP), and the angle:
tan(9°) = QR / QP
Since we know QR = h meters and the horizontal distance QP = d meters, we can rearrange the equation to solve for d:
d = QR / tan(9°)
(ii) To calculate the speed of the bird, we need to divide the distance covered (calculated in part (i)) by the time taken.
Let's assume that the bird covers the distance d in 18 seconds. To calculate the speed:
Speed = Distance / Time = d meters / 18 seconds
To convert the speed from meters per second to kilometers per hour, we need to multiply by a conversion factor:
Speed (km/h) = Speed (m/s) * 3.6
Calculate the speed using the distance calculated in part (i), and you'll have the answer.