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Airplane leaves City A and flies 142 km due north to City B. It then turns through an angle of 57° and flies 108 km to City C. What angle 8 with respect to

due north could the pilot have used to fly directly from City A to City C? See the figure below.
City
108 km
Carry your intermediate computations to at least four decimal places.
Round your answer to the nearest tenth of a degree.

Airplane leaves City A and flies 142 km due north to City B. It then turns through-example-1

1 Answer

9 votes

Answer:

24.3°

Explanation:

A suitable calculator can add the vectors directly. (See attached.) In this calculation, bearing angles are measured clockwise from north, so the initial vector has an associated angle of 0 degrees.

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We can find the angle of interest by considering the distances north and east that city C is from city B. The relevant trig functions are ...

Sin = Opposite/Hypotenuse

Cos = Adjacent/Hypotenuse

Tan = Opposite/Adjacent

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location of C relative to B

The direct distance from B to C is the hypotenuse of a right triangle whose right angle is at point "D" due north of B and due west of C. The distances of interest are ...

sin(57°) = DC/BC

DC = BC·sin(57°) = (108 km)sin(57°) ≈ 90.5764 km

cos(57°) = BD/BC

BD = BC·cos(57°) = (108 km)cos(57°) ≈ 58.8210 km

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angle of C from A

And the angle of interest can be found from ...

tan(θ) = DC/AD = (90.5764 km)/(142 km +58.8210 km) ≈ 0.451031

θ = arctan(0.451031) ≈ 24.3°

The pilot could have flown on a bearing of 24.3° to get to city C directly from city A.

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Additional comment

We can ignore the fact that bearing angles are measured CW from north, while conventional angles on a Cartesian plane are measured CCW from "east" (+x). Each coordinate system can be mapped to the other by flipping the map over along a NE-SW line (y=x), aligning the +x axis with north and the +y axis with east.

The diagram we used in working this problem is shown in the second attachment.

Airplane leaves City A and flies 142 km due north to City B. It then turns through-example-1
Airplane leaves City A and flies 142 km due north to City B. It then turns through-example-2
User WheretheresaWill
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