Another trigonometry question?

Question

The original height of the leaning tower of Pisa was 184.5 ft. At a distance of 140 ft from the base of the tower, the angle of elevation from the ground to the top of the tower is 59 degrees. How far is the tower leaning from the original vertical position?

Answer 1

You need to specify which way the tower is leaning. To solve this, let's assume it is leaning directly toward the observer.

The horizontal distance from the base of the tower to the observer is 140 ft. The horizontal distance from the top of the tower to the observer is therefore less than 140 ft. We now have a triangle with base 140 ft, tower side 184.5 feet, and the angle opposite the tower side 59°. Let the angle opposite the tower side be x. Then by the Law of Sines we have:

184.5/sin59° = 140/sinx
sinx = (140/184.5)sin59°
x = arcsin{(140/184.5)sin59°} ≈ 40.57°

The remaining angle of the triangle α is calculated as

α = 180 - 59 - x

The angle that the tower leans from vertical β is calculated as

β = 90 - α = 90 - (180 - 59 - x) = x - 31° ≈ 40.57° - 31° = 9.57°

Answer 2

Firstly, you should assume that you are standing in a position where the tower is leaning directly towards you or away from you. Otherwise you won't be able to solve this.

Also assume that your eye is at ground level. Perhaps you are lying on the ground with the side of your head on the ground. Lol.

Draw the triangle: you have three sides. One is the ground which is horizontal. The second is the tower. You know the length of each of these. The third side is the line from your eye to the top of the tower.

Now write in any angles you know. You've only been given one.

Use the Sine Rule to solve the puzzle. This will give you the angle at the top between the tower and the line of sight.

Now use the fact that there are 180 degrees in a triangle to find the angle between the tower and the ground.

Finally, find the lean by comparing this with 90 degrees which is what the angle should be. If the angle at the base it 85 degrees or 95 degrees, for example, then ethere is a 5 degree lean.