A motorist travels from a town P on a bearing of 150° to a town Q 40km away. From Q he travels to a town R on a bearing of 210° if R is located directly south of P how far is R From P
Draw the diagram, then use the law of cosines to find PR.
Answer
To find the distance between towns P and R, we need to break down the problem into steps:
Step 1: Draw a diagram
Draw a diagram to visualize the situation. Let's assume that P is the starting town, Q is the destination town after traveling 40km on a bearing of 150°, and R is the final town located directly south of P.
R
/
/
/ 40km
/
/
P---Q
Step 2: Calculate the angle between P and R
Since R is located directly south of P, the bearing from P to R would be 180°. This is because south is directly opposite to north, which is at a bearing of 0°.
Step 3: Calculate the angle between Q, R, and the line segment QR
To find the angle between Q and R, we need to subtract the bearing from Q to R (210°) from the bearing from Q to the line segment QR (180°). Therefore, the angle between Q, R, and the line segment QR is:
180° - 210° = -30°
Step 4: Use the Law of Cosines to find the distance between P and R
In a triangle, 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 our case, we know the side a (40km), side b (distance between P and R), and angle C (-30°). Therefore, we can rearrange the formula to solve for b:
b^2 = a^2 + c^2 - 2ac * cos(C)
b^2 = 40^2 + (40)^2 - 2(40)(40) * cos(-30°)
Please note that cos(-30°) = cos(30°) = sqrt(3)/2. Simplifying further:
b^2 = 1600 + 1600 - 2(1600)(sqrt(3)/2)
b^2 = 3200 - 3200(sqrt(3))
b^2 = 3200(1 - sqrt(3))
Taking the square root of both sides:
b ≈ sqrt(3200(1 - sqrt(3)))
Therefore, the distance between town P and town R is approximately sqrt(3200(1 - sqrt(3))).
To find the distance between towns P and R, we need to break down the problem into smaller steps.
Step 1: Determine the direction from Q to R
Given that R is located directly south of P, we can infer that the direction from Q to R is opposite to the bearing of 210°. Thus, the direction from Q to R is 210° + 180° = 390°. However, since angles are generally measured from 0° to 360°, we need to bring this angle back into that range. Therefore, we subtract 360° from 390° to get 30°.
Step 2: Analyze the triangle formed by P, Q, and R
By drawing a diagram, we see that we have a triangle with P, Q, and R as vertices. The side PQ has a length of 40 km. The angle at Q is 30° (found in step 1).
Step 3: Use trigonometry to find the distance between P and R
Let PR be the distance between P and R. We can use the law of cosines to find PR.
The law of cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, a = 40 km (PQ), b = PR, and C = 30°.
PR^2 = 40^2 + PR^2 - 2 * 40 * PR * cos(30°)
Simplifying the equation:
0 = 40^2 - 2 * 40 * PR * cos(30°)
Solving for PR:
2 * 40 * PR * cos(30°) = 40^2
PR * cos(30°) = 40^2 / (2 * 40)
PR * cos(30°) = 40 / 2
PR * cos(30°) = 20
PR = 20 / cos(30°)
Using a calculator, we find that cos(30°) ≈ 0.866.
PR = 20 / 0.866
PR ≈ 23.09 km
Therefore, town R is approximately 23.09 km away from town P.