Applying trigonometry to real-world problems: towers, treasure, and projectiles

Precalculus II

Leaning Tower of Pisa

The Leaning Tower of Pisa was originally 184.5 ft high. but, currently, the tower is leaning slightly from its original perpendicular state. At a distance of 123 feet from the base of the tower, the angle of elevation to the top of the tower is found to be 60°. Find the approximate angle that the tower is leaning from its original perpendicular state. Then, find the towers current height.

The current height of the tower is BD in the scratch. We can find BD from triangle ABD:

BD=AB*sin

We can find AB using cosine theorem:

BC^2= AC^2+AB^2- 2*AB*AC*cos

BC=184.5

AC=123 cos

25=15129+ AB^2 -2*AB*123*0.5

AB^2-123*AB-18911.25=0 solving this equation we should find only positive root:

D=123^2+4*18911.25

D=90774

D=301.287

AB=(123+301.287)/2

AB=212 sin

A=0.866

BD= 212*0.866

BD=183.59

Answer: tower's height is 183.59 ft.

Part II: Lost Treasure

While scuba diving in Bermuda, you discover a treasure map in a pirate schooner. The map directed them to an area that was no longer there. So, they had to re-create the map by treasure map experts.

The directions on the map read as follows:

From the tallest palm tree, sight the highest hill. Drop your eyes vertically until you sight the base of the hill.

Turn 40° clockwise from that line and walk 70 paces to the big red rock.

From the red rock, walk 50 paces back to the sight line between the palm tree and the hill. Dig there for the treasure.

Mission: Draw the scenario and show the location of your treasure. How many paces is the treasure from the palm tree?

Here's the catch! When you went to dig the treasure, it wasn't there. What happened? Where is the treasure?

SOLUTION:

In the triangle ABC, BC can be found as:

BC=AB* sin

A

BC=70*0.6427

BC?45

AC=?(AB^2-BC^2)

AC=53.6

DC can be found from triangle BCD:

DC=?(DB^2-BC^2)

DC=21.8

AD=DC+AC

AD=75.4

EA can be found from triangle ACE:

EA=?(AC^2+CE^2)

EA?54

Answer: treasure can be either in point E. which is 54 paces from palm tree or in point D. which is 75.4 paces from the palm tree

Part IV: Playing Catch

You are playing catch with a friend located 100 feet away. If you throw the ball with an initial velocity of vo = 90 feet per second, at what approximate angle of elevation ? should you direct your throw so that it can be caught easily? Use the formula, where'd is the horizontal distance that the ball travels. Is there more than one answer?

Solution:

knowing that sin? =cos (0.5?-?) we can find that another angle at which ball can travel equals to Answer: equals to 78.37 or 11.63 degrees

Precalculus II

60°

123 ft