Bobby is standing near a lighthouse. He measured the angle formed from where he stood to the top of the lighthouse and it was 30 degrees. Then he backed up 40 feet and measured the angle again and it was 25 degrees. Solve the height of the lighthouse.

Answer 1

Label the original position as A and the second position as B, top of the lighhouse as P and its base as Q. ABQ is a straight line, AB = 40, angle B=25°, angle PAQ = 30°, AQ = x, and PQ = h

I used to teach this question is 2 ways:

1.
In triangle PBA, angle PAB = 150°
then angle BPA = 5°
by the sine law:
PA/sin25 = 40/sin5°
PA = 40sin25/sin5

in the right-angled triangle PAQ,
sin30° = h/PA
h = PAsin30 = (40sin25/sin5)(sin30) = 96.98..

method2:
in triangle PAQ, tan30 = h/x
h = xtan30
in triangle PBQ, tan25 = h/(x+40)
h = (x+40)tan25

thus: xtan30 = (x+40)tan25
xtan30 = xtan25 + 40tan25
x(tan30 - tan25) = 40tan25
x = 40tan25/(tan30-tan25)

h = xtan30
= (40tan25)(tan30)/(tan30-tan25)
= 96.98..

I consider method 1 as the easier of the two

Answer 2

Or, method two simplifies a bit if you're not afraid of the cot(x) function:

h cot25 - h cot30 = 40

Answer 3

I also got the answer of 96.98 ft, but the answer key says 18.4 ft. Who is right?

Answer 4

If the answer key says 18.4 ft, it's possible that they made a mistake or gave you a different problem. Solving the problem using the given information should lead to a height greater than 18.4 ft.