A lighthouse standing on the top of a cliff is observed from two open boats (boats are not decked over) A and B in a vertical plane through the light house. The angle of elevation of the top of the lighthouse viewed from B is 16 degrees and the angles of elevation of the top and bottom viewed from A are 40 degrees and 23 degrees respectively. If the boats are 1320 ft. apart, find the height of the lighthouse and the height of the cliff.
c = cliff height
h = lighthouse hight
a = distance of A from cliff
b = distance of B from cliff
c/a = tan 23°
(c+h)/a = tan 40°
(c+h)/b = tan 17°
So, clearing fractions and plugging in the trig values, we have
c = .4245a
c+h = .8391a
c+h = .3057b
b = a+1320
h = 313.65
c = 321.14
To find the height of the lighthouse and the height of the cliff, we can use trigonometric ratios and the information given in the problem.
Let's denote the height of the lighthouse as 'h' and the height of the cliff as 'x'. We can break down the problem into two separate triangles: Triangle A-Lighthouse and Triangle B-Lighthouse.
1. Triangle A-Lighthouse:
From Triangle A-Lighthouse, we have the following information:
- Angle of elevation of the top of the lighthouse from A = 40 degrees
- Angle of elevation of the bottom of the lighthouse from A = 23 degrees
Using these angles, we can find the height of the lighthouse, h.
We can use the trigonometric ratio tangent (tan):
tan(angle) = opposite/adjacent
Using the angle of 40 degrees, we have:
tan(40 degrees) = h/x
Solving for h, we get:
h = x * tan(40 degrees)
Next, we can use the angle of 23 degrees to find the height of the bottom of the lighthouse from A:
tan(23 degrees) = (h - x)/x
Simplifying this equation, we get:
x * tan(23 degrees) = h - x
2. Triangle B-Lighthouse:
From Triangle B-Lighthouse, we have the following information:
- Angle of elevation of the top of the lighthouse from B = 16 degrees
Using this angle, we can again use the tangent ratio to find the height of the lighthouse, h.
tan(16 degrees) = h/(x+1320)
Simplifying this equation, we get:
h = (x + 1320) * tan(16 degrees)
Now, we have two equations for the height of the lighthouse, h, based on both Triangle A-Lighthouse and Triangle B-Lighthouse. We can set them equal to each other and solve for x:
x * tan(40 degrees) = (x + 1320) * tan(16 degrees)
Once we find the value of x, we can substitute it back into either equation to find the height of the lighthouse, h.