A child on top of a lighthouse observes two ships that are 90 ft apart from each other. The angles of the depression of the two ships are 34º and 20º. How far is the closest ship from the base of the lighthouse?
I'm sorry, I would show my work, I just don't know if I could put a picture on here.
Let's assume that they are in the same line of sight, that is, the two ships and the lighthouse are in the same plane.
Label the farther ship A and the nearer ship B
Label the top of the lighthouse L and its bottom M
I see the following angles,
angle A = 20°
angle LBM =34° and AB=90 ft
in triangle LAB
angle A= 20
angle ABL = 180-34 = 146° which makes
angle ALB = 14°
by the Sine Law:
LB/sin20 = 90/sin14
LB = 90sin20/sin14
In the right-angled triangle LBM,
BM/LB= cos34
BM = LBcos34
= (90sin20/sin14)(cos34)
= ....
only now would I go to my calculator
Many people would do this using tangents or cotangents, but I always think that way is harder to understand.
No problem! I can help you solve this problem step by step.
To find the distance from the closest ship to the base of the lighthouse, we can use trigonometry. Specifically, we can use the tangent function.
Let's assume that the closest ship is Ship A and the other ship is Ship B. The angles of depression can be represented as follows:
Angle of Depression of Ship A = 34º
Angle of Depression of Ship B = 20º
Now, let's draw a diagram to better visualize the situation. Make a right triangle with the lighthouse at the top and the base of the lighthouse at the bottom. The two ships can be placed on either side of the lighthouse at different heights.
To determine the distance from the closest ship (Ship A) to the base of the lighthouse, we need to find the length of the opposite side of the right triangle.
Using the tangent function, we can write:
tan(angle of depression) = opposite / adjacent
For Ship A:
tan(34º) = opposite distance / distance between ships (90 ft)
Now, we can use trigonometric properties to find the distance from Ship A to the lighthouse's base.
First, solve for the opposite distance:
opposite distance = tan(34º) * 90 ft
Using a calculator, we find:
opposite distance ≈ 68.05 ft
Therefore, the closest ship (Ship A) is approximately 68.05 ft away from the base of the lighthouse.