A treasure map directs you to start at palm tree
and walk due north for 15.0 m. You are then to turn
90° and walk 22.0 m; then turn 90° again and
walk 5.00 m.
Give the distance from the palm tree,and the direction to north, for each of the four possible locations of the treasure .

The direction of the 90° turns are the possible directions used for the calculations

The four distances and directions are;

Reason:

Let point A represent the motion of the treasure hunter, we have:

Turning West then West;

Walking due north to location (0, 15)

Turn 90° West and walk 22.0 m to the location (-22, 15)

Turn 90° West again and walk 5.00 m. to the location (-22, 10)

The location of point A = (-22, 10)

[tex]Direction \ of \ point \ A = arctan \left(\dfrac{10}{-22} \right) \approx -24.4^{\circ}[/tex]

Direction to North = 90° - 24.4° ≈ 65.6°

Distance = √((-22)² + 10²) ≈ 24.2

Therefore, we have;

Turning East then West:

Turn 90° East and walk 22.0 m to the location (22, 15)

Turn 90° West again and walk 5.00 m. to the location (22, 20)

The location of point A = (22, 20)

[tex]Direction \ of \ point \ A = arctan \left(\dfrac{20}{22} \right) \approx 42.7^{\circ}[/tex]

Direction to North = 90° - 42.3° ≈ 47.7° East

Distance = √((-22)² + 20²) ≈ 29.7

Therefore, we have;

Turning East then East:

Turn 90° East and walk 22.0 m to the location (22, 15)

Turn 90° East again and walk 5.00 m. to the location (22, 10)

The location of point A = (22, 10)

[tex]Direction \ of \ point \ A = arctan \left(\dfrac{10}{22} \right) \approx 24.4^{\circ}[/tex]

Direction to North = 90° - 24.4° ≈ 65.6° East

Distance = √((-22)² + 10²) ≈ 24.2

Therefore, we have;

Turning West then East:

Turn 90° West and walk 22.0 m to the location (-22, 15)

Turn 90° East and walk 5.00 m. to the location (-22, 20)

The location of point A = (-22, 20)

[tex]Direction \ of \ point \ A = arctan \left(\dfrac{20}{-22} \right) \approx -42.7^{\circ}[/tex]

Direction to North = 90° - 42.7° ≈ 47.7° West

Distance = √((-22)² + 20²) ≈ 29.7

Therefore, we have;

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