Q is 80km from p. r is 100km from q an a bearing of 066° . Angle pqr equals 054°
Calculate the bearing of p from q
The distance pr correct to 2 d.p
The measurement of angle qpr to the nearest degree.
P from Q is 66-54 = 12°
PR^2 = 80^2 + 100^2 - 2*80*100 cos54°
sinP/100 = sin54°/PR
35
To calculate the bearing of p from q, we need to subtract the angle pqr from 180 degrees since the bearing is measured clockwise from the north.
1. Calculate the bearing of p from q:
Bearing of p from q = 180° - angle pqr
Bearing of p from q = 180° - 54°
Bearing of p from q = 126°
2. To find the distance pr, we can use the cosine rule since we have two sides and an angle opposite one of the sides.
Distance pr = √(qp^2 + qr^2 - 2qp * qr * cos(angle pqr))
Distance pr = √(80^2 + 100^2 - 2*80*100*cos(54°))
Distance pr = √(6400 + 10000 - 16000*cos(54°))
Distance pr = √(6400 + 10000 - 16000*(-0.58778525))
Distance pr = √(6400 + 10000 + 9383.2552)
Distance pr = √(2583.2552)
Distance pr ≈ 50.82 km (to 2 decimal places)
3. The measurement of angle qpr can be calculated by subtracting the angle qrp from 180 degrees.
Angle qpr = 180° - angle qrp
Angle qpr = 180° - 66°
Angle qpr = 114°
Therefore, the answers are:
- The bearing of p from q is 126°.
- The distance pr is approximately 50.82 km (to 2 decimal places).
- The measurement of angle qpr is approximately 114°.
To calculate the bearing of point P from Q, we need to subtract the angle PQR from the bearing of Q from R.
1. Bearing of Q from R: Since the bearing of Q from R is given as 066°, we can use this value directly.
2. Angle PQR: The angle PQR is given as 054°, so we can use this value directly as well.
To calculate the bearing of P from Q:
3. Subtract the angle PQR from the bearing of Q from R:
Bearing of P from Q = Bearing of Q from R - Angle PQR
Bearing of P from Q = 066° - 054°
Bearing of P from Q = 012°
Therefore, the bearing of P from Q is 012°.
To calculate the distance PR correct to 2 decimal places:
4. Use the Cosine Rule:
PR^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(angle QPR)
Given:
PQ = 80 km
QR = 100 km
Angle QPR = 54° (as provided, but we'll convert it to radians)
Angle QPR in radians = 54° * (π/180)
Angle QPR in radians = 0.94248 radians
PR^2 = 80^2 + 100^2 - 2 * 80 * 100 * cos(0.94248)
PR^2 = 6400 + 10000 - 2 * 80 * 100 * cos(0.94248)
PR^2 = 14400 - 16000 * cos(0.94248)
PR^2 ≈ 14400 - 16000 * (0.6249) (cosine value rounded to 4 decimal places)
PR^2 ≈ 14400 - 9988
PR^2 ≈ 4412
PR ≈ √4412
PR ≈ 66.42 km
Hence, the distance PR correct to 2 decimal places is approximately 66.42 km.
To calculate the measurement of angle QPR to the nearest degree:
5. Use the Sine Rule:
sin(angle QPR) = (PR * sin(angle PQR)) / QR
Given:
PR = 66.42 km (as calculated above)
QR = 100 km
Angle PQR = 54° (as provided, we'll convert it to radians)
Angle QPR = the angle we want to find
Angle PQR in radians = 54° * (π/180)
Angle PQR in radians = 0.94248 radians
sin(angle QPR) = (66.42 * sin(0.94248)) / 100
sin(angle QPR) ≈ (66.42 * 0.8165) / 100 (sin value rounded to 4 decimal places)
sin(angle QPR) ≈ 54.1923 / 100
sin(angle QPR) ≈ 0.5419
angle QPR ≈ arcsin(0.5419) * (180/π)
angle QPR ≈ 31.2° (rounded to the nearest degree)
Therefore, the measurement of angle QPR to the nearest degree is approximately 31°.