Chapter 5.7, Problem 35E

To determine

How far is the tower leaning from the original vertical position?

Answer to Problem 35E

  $\text{leaning }{9.6}^{\circ }\text{ far from the original vertical position}$

Explanation of Solution

Given information: Architecture: The original height of the Leaning Tower of Pisa was $184\frac{1}{2}$ feet. At a distance of 140 feet from the base of the tower, the angle of elevation from the ground to the top of the tower is ${59}^{\circ }$ .

Calculation:

The angle at the base was originally ${90}^{\circ }$ . Let us see what it is now.

Use the yellow triangle.

And the law of sines

Label the points C-base, B-top of tower, A-at the measured angle.

  $\begin{array}{l}\frac{184.5}{\sin {59}^{\circ }}=\frac{140}{\sin B}\\ \sin B=\frac{140\cdot \sin {59}^{\circ }}{184.5}\\ B={\sin }^{-1}\left(\frac{140\cdot \sin {59}^{\circ }}{184.5}\right)\\ B\approx 40.57365\text{ is the angle at the top of the tower}\end{array}$

At the base,

  $C\approx {180}^{\circ }-\left({59}^{\circ }+{40.6}^{\circ }\right)={80.4}^{\circ }$

So the difference is,

  ${90}^{\circ }-{80.4}^{\circ }\approx {9.6}^{\circ }$

Thus,

  $\text{leaning }{9.6}^{\circ }\text{ far from the original vertical position}$