Trigonometry Question

Question

A surveyor standing on a hill 19 feet high looks at a building across a river. The surveyor determines that the angle of depression to the base of the building is 20.23 degrees and the angle of elevation to the top of the building to be 39.383 degrees. Calculate the height of the building in feet to two decimal places.

Answer 1

Draw and label a diagram!

H is is the top of the hill

F is the bottom of the hill

T is the top of the building

M is the point on the building directly opposite H

B is base of he building.

TB = TM + MB

BM = FH

BM = 19

BF/ 19 = tan 69.77°

Determine BF

HM = BF

tan 39.383° = TM / HM

Answer 2

Not having any additional information, I will assume that the base of the building is at the same horizontal elevation as the base of the hill. The angle of depression would be measured from the horizontal line of sight of the surveyor. From a drawing of the situation, the height of the building from the base to the horizontal line of sight of the surveyor should be 19 ft.

To calculate the height of the upper part of the building, you need to know the length of one side of the triangle made by horizontal line of sight of the surveyor, the height of the upper part of the building and the line of sight to the top of the building. You can calculate the horizontal line of sight distance to the building by the following equation: tan 20.23 = 19/D where D is the horizontal line of sight distance from the surveyor to the building. D = 51.56 feet.

From this you can calculate the height of the upper part of the building using tan 39.383 = x/51.56. Solve for x to get x = 51.56 x tan 39.383 = 42.34 feet. The total height of the building is 19 + 42.34 = 61.34 feet.

This answer depends upon the correct interpretation of the geometry of the problem. An included drawing would have been helpful.