Precalculus D 9.2 Application Examples

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: http://en.wikipedia.org/wiki/Leaning_Tower_of_Pisa

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

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

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

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

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

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

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

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

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”

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.

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

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

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.

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.

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

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.

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.

Airplane Height Start by solving for the angles

Airplane Height Start by solving for the angles

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

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

Airplane Height Now solve for h using SOH-CAH-TOA Thus the airplane is 710.97’ high