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?
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
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.