Question

the bearing from city a to city b is n 49° e. the bearing from city b to city c is s 41° e. an automobile driven at 55 mph takes 1.6 hours to drive from city a to city b and takes 1.4 hours to drive from city b to city c. find the distance from city a to city c. (neglect the curvature of the earth.)

Explanation:

Step1: Calculate distance from A to B

Distance = speed × time. Speed is 55 mph, time from A to B is 1.6 hours.
$d_{AB} = 55 \times 1.6 = 88$ miles.

Step2: Calculate distance from B to C

Time from B to C is 1.4 hours, speed 55 mph.
$d_{BC} = 55 \times 1.4 = 77$ miles.

Step3: Determine angle at B

Bearing from A to B is N 49° E, from B to C is S 41° E. The angle between AB and BC: 49° + 41° = 90°, so triangle ABC is right - angled at B.

Step4: Apply Pythagorean theorem

In right - triangle ABC, $d_{AC}=\sqrt{d_{AB}^{2}+d_{BC}^{2}}$. Substitute $d_{AB} = 88$ and $d_{BC}=77$.
$d_{AC}=\sqrt{88^{2}+77^{2}}=\sqrt{7744 + 5929}=\sqrt{13673}\approx116.93$ miles.

Answer:

The distance from City A to City C is approximately 116.93 miles.