A town p is 120km from town q on a bearing of north 60degree east town r is 76km from town p on bearing of south 50degree east. Find the bearing and distance of town q from r
angle QPR = 60 + 50 = 110 deg
law of cosines
120^2 + 76^2 = QR^2 - 2 *120 * 76 * cos 110
then
sin 110 / QR = sin PQR / 76
To find the bearing and distance of town Q from town R, we can use the concept of a triangle formed by these three towns. Let's solve it step by step:
1. Draw a diagram: Start by drawing a diagram with the given information. Draw the towns P, Q, and R as points on the paper.
2. Label the sides: Label the sides of the triangle formed by P, Q, and R. Let the side connecting P and Q be labeled as PQ, the side connecting Q and R be labeled as QR, and the side connecting R and P be labeled as RP.
3. Find the length of PQ: It is given that town P is 120 km from town Q. So, the length of PQ is 120 km.
4. Find the length of PR: The length of PR can be found by using the distance between P and R. It is given that town R is 76 km from town P. So, the length of PR is 76 km.
5. Find the angles: To find the bearing of Q from R, we need to find the angles of the triangle. To do this, use the given bearings.
a. North 60 degrees east: This means that the angle at P (formed by lines PQ and PR) is 60 degrees.
b. South 50 degrees east: This implies that the angle at R (formed by lines RQ and RP) is 50 degrees.
6. Find the angle at Q: To find the angle at Q (formed by lines QR and QP), use the fact that the sum of the angles in a triangle is 180 degrees. So, angle at Q = 180 - (angle at P + angle at R).
angle at Q = 180 - (60 + 50) = 70 degrees.
7. Use the Law of Cosines: We can now use the Law of Cosines to find the length of QR.
The Law of Cosines states that c^2 = a^2 + b^2 - 2ab * cos(C), where c is the side opposite to angle C.
In this case, a = PQ (known), b = PR (known), and C = angle at Q (known).
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(angle at Q).
QR^2 = 120^2 + 76^2 - 2 * 120 * 76 * cos(70).
QR^2 ≈ 14400 + 5776 - 18240 * 0.3420 (using a calculator to approximate the cosine).
QR^2 ≈ 14400 + 5776 - 6242.88.
QR^2 ≈ 20933.12.
Taking the square root of both sides, we get QR ≈ √20933.12 ≈ 144.6 km.
8. Find the bearing: The bearing of Q from R is the compass direction from R to Q. To find it, we need to determine the direction in degrees clockwise from the northern direction.
To calculate the bearing, use the Law of Sines. The Law of Sines states that sin(A) / a = sin(B) / b = sin(C) / c, where A, B, and C are angles and a, b, and c are the sides opposite to those angles.
sin(angle at R) / QR = sin(angle at Q) / RP.
sin(50) / QR = sin(70) / 76.
QR ≈ (sin(70) * 76) / sin(50).
QR ≈ 91.72 km.
Since we know the exact length of QR (144.6 km), we can find the exact values of the sines.
9. Therefore, the bearing of Q from R is approximately 70 degrees, and the distance from Q to R is approximately 91.72 km.
Remember to double-check the calculations and ensure the accuracy of the final answer.