from the top of a building a person spots two boats out on the lake. the angle of depression from the person to one of the boats in 54 degrees, the angle of depression from the person to the other boat is 31 degrees. how far apart are the boats if the person is 35 meters above the ground?

Distance between the two boats = 32.82m

Step-by-step explanation:

Since the person is standing on top of the building and seeing 2 boats on the same side of him, we have two right triangles.

However, instead of the two triangles being adjacent to each other and therefore additive, one is inside the other and thus we will subtract to get the distance between the two boats.

Since we know the vertical height of the building above the ground instead of the horizontal distance distance to the boats, we will be dividing instead of multiplying.

Thus, using trigonometric ratios, the boat with the bigger angle of depression would be nearer to the base of the building than the other one which is boat 2. Hence, we have;

Horizontal distance of Boat one to base of building:

tan54 = 35/distance

Rearranging;

distance = 35/tan (54)

distance = 35/1.3764

distance = 25.43m

Horizontal distance of boat two to base of building:

tan (31) = 35/distance

distance = 35/tan (31)

distance = 35/0.6009

distance = 58.25m

so distance between the two boats = 58.25 - 25.43 = 32.82m