Leaning Tower of Pisa

[greatwhiteshark]

The famous Leaning Tower of PIsa was originally 184.5 feet high. At a distance 123 feet from the base of the tower, the angle of elevation to the top of the tower is 60 degrees. Find angle CAB and find the perpendicular distance from angle C to side AB.

This is also a Law of Sines question. I drew a picture of this question but don't know how to draw it using a keyboard.

 

[soroban]

Hello, greatwhiteshark!

Even if you can't draw a diagram, a clear description would help.
For example, where is A? ... on the ground? at the top of the tower?

The famous Leaning Tower of PIsa was originally 184.5 feet high.
At a distance 123 feet from the base of the tower, the angle of elevation to the top of the tower is 60^o.

Find angle CAB and find the perpendicular distance from angle C to side AB.

                C
                 *             The tower is CA = 184.5 feet.
               / |\  
             /   | \              Angle B = 60 degrees.
           /     |  \184.5
         /       |   \                BA = 123 feet 
       /60       |    \
    B* - - - - - + - - *A
                 D
     ------- 123 -------

. . . . . . . . . . . . sin C . . . .sin B
Law of Sines: . ------ . = . ------
. . . . . . . . . . . . . BA . . . . . CA

. . . . . . . . . .sin C . . . . sin 60^o
We have: . ------- . = . ---------
. . . . . . . . . .123 . . . . . 184.5

. . . Then: . sin C[/color=yellow] .[/color]= .0.577350269

. . . . . . . . . . . .C . ≈ . 35.26^o

Therefore: . <u>/</u>CAB .= .180^o - 60^o - 35.26^o .= .84.74^o


Then: . AD .= .184.5 sin 84.74^o .≈ .183.72 feet

 

[greatwhiteshark]

Good

Yes, C is at the top and all the letters at the bottom.

 

[MordecaiBleck]

By the time builders had finished the third of eight planned stories about five years later, the tower's foundation had begun to settle unevenly on the ground beneath it, a dense mixture of clay, sand and shells. As a result, the structure had begun to tilt visibly toward the south

Why does the Leaning Tower of Pisa not fall?


The Center of Gravity

In short, this is why the Tower of Pisa does not topple over. The Leaning Tower doesn't fall because its center of gravity is carefully kept within its base

 

[blamocur]

Hello, greatwhiteshark!

Even if you can't draw a diagram, a clear description would help.
For example, where is A? ... on the ground? at the top of the tower?

                C
                 *             The tower is CA = 184.5 feet.
               / |\ 
             /   | \              Angle B = 60 degrees.
           /     |  \184.5
         /       |   \                BA = 123 feet
       /60       |    \
    B* - - - - - + - - *A
                 D
     ------- 123 -------

. . . . . . . . . . . . sin C . . . .sin B
Law of Sines: . ------ . = . ------
. . . . . . . . . . . . . BA . . . . . CA

. . . . . . . . . .sin C . . . . sin 60^o
We have: . ------- . = . ---------
. . . . . . . . . .123 . . . . . 184.5

. . . Then: . sin C[/color=yellow] .[/color]= .0.577350269

. . . . . . . . . . . .C . ≈ . 35.26^o

Therefore: . <u>/</u>CAB .= .180^o - 60^o - 35.26^o .= .84.74^o


Then: . AD .= .184.5 sin 84.74^o .≈ .183.72 feet

Nitpicking now being my middle name I want to point out that the answer depends on the direction between DB and DA. E.g., the answer would be very different if A were between D and B.