To determine how far Philip walked altogether, we first need to break down the journey into individual legs and then calculate the total distance.
1. Distance Walked from P to Q:
Philip starts at point P and walks 50 meters to point Q on a bearing of 070°.
Therefore, the distance [tex]\( PQ \)[/tex] is 50 meters.
2. Distance Walked from Q to R:
From Q, Philip continues to walk 80 meters to point R on a bearing of 120°.
Thus, the distance [tex]\( QR \)[/tex] is 80 meters.
3. Calculating Coordinates of Point Q:
- The bearing of 070° means that point Q is positioned at an angle of 70° from the North.
- Breaking this down into x and y coordinates using trigonometry:
[tex]\[
x_Q = 50 \times \sin(70°)
\][/tex]
[tex]\[
y_Q = 50 \times \cos(70°)
\][/tex]
4. Calculating Coordinates of Point R:
- From point Q, moving towards R on a bearing of 120° means this is at [tex]\(120° - 70° = 50°\)[/tex] from the line PQ.
- We calculate the new coordinates:
[tex]\[
x_R = x_Q + (80 \times \sin(120°))
\][/tex]
[tex]\[
y_R = y_Q + (80 \times \cos(120°))
\][/tex]
5. Distance Walked from R to P:
- Having the coordinates of R, we can compute the straight-line distance from R back to P. This is done using the Pythagorean theorem:
[tex]\[
RP = \sqrt{x_R^2 + y_R^2}
\][/tex]
6. Total Distance Walked:
- Finally, to find the total distance Philip walked, we sum up the distances PQ, QR, and RP:
[tex]\[
\text{Total distance} = PQ + QR + RP
\][/tex]
From our calculations, we find the following distances:
- [tex]\( PQ = 50 \)[/tex] meters
- [tex]\( QR = 80 \)[/tex] meters
- [tex]\( RP \approx 118.50 \)[/tex] meters
Therefore, the total distance that Philip walked is approximately:
[tex]\[
50 + 80 + 118.50 = 248.50 \text{ meters}
\][/tex]
Thus, the total distance Philip walked altogether is approximately 248.50 meters.
