Question

While sailing a boat offshore, Donna sees a lighthouse and calculates that the angle of elevation to the top of the lighthouse is 3 degrees, as shown in the accompanying diagram. When she sails her boat 700 feet closer to the lighthouse, she finds that the angle of elevation is now 5 degrees. How tall, to the nearest tenth of a foot, is the lighthouse?

Please show all work on how you got your answer

Answer 1

Answer: the height of the lighthouse is 91.4 feet.

Explanation:

Let h represent the height of the lighthouse.

Let x represent the base of the smaller right angle triangle so that the length of the base of the big right triangle would be (x + 700) feet.

Considering the bigger right angle triangle,

Tan θ = opposite side/adjacent side

Tan 3 = h/(x + 70)

0.0524 = h/(x + 700)

h = 0.0524(x + 70)

h = 0.0524x + 36.668 - - - - - - - - -1

Considering the smaller right angle triangle,

Tan 5 = hx

h = xtan5 = x × 0.0875

h = 0.0875x - - - - - - - - -2

Equating equation 1 to equation 2, it becomes

0.0524x + 36.668 = 0.0875x

0.0875x - 0.0524x = 36.668

0.0351x = 36.668

x = 36.668/0.0351

x = 1044.67

h = 0.0875x = 0.0875× 1044.67

h = 91.4