The boundaries PQ, QR, RS and SP of a ranch are straight lines such that: Q is 16km on a bearing of 040° from P, R is directly south of Q and east of P and S is 12km on a bearing of 120° from R.
(b) From the scale drawing determine:
(i) The distance, in kilometers, of P from S
(ii) The bearing bf P from S
(c) Calculate the area of the ranch PQRS in square kilometers.
Did you make a drawing?
PR = 16 cos50° = 10.28
(i) PS^2 = PR^2 + 12^2 - 2*PR*12*cos150°
PS = 21.53
(ii) In triangle PRS, angle P is found by
sinP/12 = sin150°/21.53
The bearing of S from P is 90°+angle P
Thanks
To solve this problem, let's break it down step by step.
Step 1: Find the coordinates of each point
First, let's find the coordinates of each point based on the given information.
P is the starting point, so we can assign it the coordinates (0, 0). The bearing of 040° means that Q is 16km away from P in the direction of 40° from the positive x-axis. Using trigonometry, we can calculate the coordinates of Q.
To find the x-coordinate of Q, we can use cosine:
x-coordinate of Q = x-coordinate of P + 16km * cos(40°)
= 0 + 16km * cos(40°)
≈ 12.256 km
To find the y-coordinate of Q, we can use sine:
y-coordinate of Q = y-coordinate of P + 16km * sin(40°)
= 0 + 16km * sin(40°)
≈ 10.264 km
So, the coordinates of Q are approximately (12.256 km, 10.264 km).
Since R is directly south of Q, its x-coordinate will be the same as that of Q. To find the y-coordinate of R, we need to subtract the distance from Q to R, which is not given. However, we can calculate it using trigonometry. Since R is directly south of Q, it forms a right triangle with Q and P. The bearing of 40° from P to Q means that the bearing of 90° from P to R.
Based on this, we can calculate the distance from Q to R using Pythagoras' theorem:
Distance from Q to R = sqrt((x-coordinate of Q - x-coordinate of P)^2
+ (y-coordinate of Q - y-coordinate of P)^2)
= sqrt((12.256 km - 0)^2 + (10.264 km - 0)^2)
= sqrt(150.339 km^2)
≈ 12.266 km
Therefore, the y-coordinate of R is approximately (10.264 km - 12.266 km) = -2.002 km.
So, the coordinates of R are approximately (12.256 km, -2.002 km).
Finally, to find the coordinates of S, we can use the distance of 12km and the bearing of 120° from R. We can calculate the x-coordinate and y-coordinate of S using trigonometry.
To find the x-coordinate of S, we can use cosine:
x-coordinate of S = x-coordinate of R + 12km * cos(120°)
= 12.256 km + 12km * cos(120°)
≈ 3.778 km
To find the y-coordinate of S, we can use sine:
y-coordinate of S = y-coordinate of R + 12km * sin(120°)
= -2.002 km + 12km * sin(120°)
≈ -16.834 km
So, the coordinates of S are approximately (3.778 km, -16.834 km).
Step 2: Find the distance of P from S
To find the distance of P from S, we can use the distance formula:
Distance between two points = sqrt((x-coordinate of S - x-coordinate of P)^2
+ (y-coordinate of S - y-coordinate of P)^2)
= sqrt((3.778 km - 0)^2 + (-16.834 km - 0)^2)
= sqrt(302.271 km^2)
≈ 17.391 km
Therefore, the distance of P from S is approximately 17.391 km.
Step 3: Find the bearing of P from S
To find the bearing of P from S, we can calculate the angle between the line connecting P and S and the positive x-axis.
Using trigonometry, the bearing can be found using arctan:
Bearing of P from S = arctan((y-coordinate of P - y-coordinate of S)
/ (x-coordinate of S - x-coordinate of P))
Bearing of P from S = arctan((0 - (-16.834 km)) / (3.778 km - 0))
= arctan(16.834 km / 3.778 km)
≈ 77.491°
Therefore, the bearing of P from S is approximately 77.491°.
Step 4: Find the area of the ranch PQRS
To find the area of the ranch PQRS, we need to divide it into two triangles: PQR and PSR.
The area of a triangle can be calculated using the formula:
Area of triangle = 1/2 * base * height
For the triangle PQR, the base is the distance between P and Q, and the height is the distance from Q to the line PR (which is the y-coordinate of R):
Area of triangle PQR = 1/2 * distance from P to Q * y-coordinate of R
= 1/2 * 16km * (-2.002 km)
≈ -16.032 km^2
Keep in mind that the area value is negative because the triangle is oriented downward.
For the triangle PSR, the base is the distance between P and S, and the height is the distance from S to the line PR (which is the y-coordinate of S):
Area of triangle PSR = 1/2 * distance from P to S * y-coordinate of S
= 1/2 * 17.391 km * (-16.834 km)
≈ -145.519 km^2
Similarly, the area value is negative due to the downward orientation.
Finally, to find the area of the ranch PQRS, we add the absolute values of the two triangle areas:
Area of ranch PQRS = |Area of triangle PQR| + |Area of triangle PSR|
= |-16.032 km^2| + |-145.519 km^2|
= 16.032 km^2 + 145.519 km^2
≈ 161.55 km^2
Therefore, the area of the ranch PQRS is approximately 161.55 square kilometers.