1. Precalculus D
9.2 Application Examples

2. Height of the Leaning Tower of Pisa
The tower was built in 1173 and had a poor foundation laid on soil that settled
The tower was originally 183’8” high
If you move 141’ away from the base (opposite the lean), the angle of elevation is 50°
What angle is the tower tilted from vertical?
What is the current height of the tower?
Source:

3. Height of the Leaning Tower of Pisa
Draw a triangle with the given information
We will solve for Angle A eventually
To get to Angle A we must first solve for Angle C

4. Height of the Leaning Tower of Pisa
Set up Law of Sines then solve for C:

5. Height of the Leaning Tower of Pisa
Solve for Angle A:

6. Height of the Leaning Tower of Pisa
Solve for Angle A:
Thus the tower is leaning 3.98° from perfectly vertical

7. Height of the Leaning Tower of Pisa
Solve for Angle A:
Thus the tower is leaning 3.98° from perfectly vertical

8. Height of the Leaning Tower of Pisa
To solve for the current vertical height set up another triangle
First find Angle A

9. Height of the Leaning Tower of Pisa
To solve for the current vertical height set up another triangle
First find Angle A

10. Height of the Leaning Tower of Pisa
Now solve for “a” using SOH-CAH-TOA

11. Height of the Leaning Tower of Pisa
Now solve for “a” using SOH-CAH-TOA
Thus the tower is 183’2.75” tall now, down about 5.25”

12. Airplane Height Two sensors are spaced 700 feet apart.
When the aircraft is at point A on its approach, the angle of elevations are:
from the first sensor (F) to the aircraft is 20°
from the second sensor (S) to the aircraft is 15°
Determine how high the aircraft is at point A.

13. Airplane Height First label the triangle with the given information
Remember that an angle of elevation is from horizontal up

14. Airplane Height First label the triangle with the given information
Remember that an angle of elevation is from horizontal up

15. Airplane Height First label the triangle with the given information
Remember that an angle of elevation is from horizontal up
We are eventually trying to find the height of the airplane, so add point B directly beneath the plane.

16. Airplane Height First label the triangle with the given information
Remember that an angle of elevation is from horizontal up
We are eventually trying to find the height of the airplane, so add point B directly beneath the plane.

17. Airplane Height
To eventually solve for h, we need to know AF, which I will call s

18. Airplane Height
To eventually solve for h, we need to know AF, which I will call s
To find s we will need to solve for all of the other angles ΔAFS, which I will call X and Y.

19. Airplane Height
To eventually solve for h, we need to know AF, which I will call s
To find s we will need to solve for all of the other angles ΔAFS, which I will call X and Y.

20. Airplane Height
Start by solving for the angles

21. Airplane Height
Start by solving for the angles

22. Airplane Height Start by solving for the angles
Now solve for s using the Law of Sines

23. Airplane Height Start by solving for the angles
Now solve for s using the Law of Sines

24. Airplane Height Now solve for h using SOH-CAH-TOA
Thus the airplane is ’ high