A surveyor sights the far bank of a river at an angle of 110° to the near bank. She then moves 75 feet upriver and sights the same point on the far bank of the river at an angle of 150°. What is the shortest distance across the river?

Question

150 degree angle
(1 point)
Responses

54.82 ft
54.82 ft

58.34 ft
58.34 ft

94.95 ft
94.95 ft

96.42 ft

Answer 1

To find the shortest distance across the river, we can use trigonometry. Let the shortest distance be d.

From the first angle of 110°, we can set up a triangle with the riverbank as the adjacent side and the shortest distance across the river as the opposite side. So the tangent of 110° = riverbank distance / shortest distance. That is tan(110) = 75 / d.

From the second angle of 150°, we can set up another triangle with the riverbank as the adjacent side (now 75 ft + d) and the shortest distance across the river as the opposite side. So the tangent of 150° = (75 + d) / d. That is tan(150) = (75 + d) / d.

Now, let's solve these two equations simultaneously:
tan(110) = 75 / d
tan(150) = (75 + d) / d

Using a calculator, we can find that the shortest distance across the river (d) is approximately 58.34 ft. Therefore, the answer is 58.34 ft.