Chapter MPS, Problem 13.6MPS

To determine

Find the height of the tower.

Answer to Problem 13.6MPS

  $183.1388feet$

Explanation of Solution

Given information:

Miguel needs to measure the height of the Leaning Tower of Pisa. He walks exactly $200$ feet from the base of the tower and looks up. The angle from the ground to the top of the tower is $44.3°$ . The leaning tower of Pisa leans about $4°$ towards Miguel.

Calculation:

The base of the triangle is $a=200feet$ and one of the angle of triangle is $B=44.3°$ .

The given figure of the triangle $\Delta ABC$ is,

  

The angle $C$ of triangle $\Delta ABC$ is as follows:

  $\begin{array}{l}\angle PCA+C=90°\\ C=90°-4°\\ C=86°\end{array}$

So, the angle $C$ of the triangle $\Delta ABC$ is $C=86°$ .

The sum of interior angles of the triangle are equal to $180°$ .

That is, $A+B+C=180°$

Use above formula and find the third angle $A$ of the triangle $\Delta ABC$ .

  $\begin{array}{l}A+44.3°+86=180°\\ A=180°-130.3°\\ A=49.7°\end{array}$

Hence, the third angle of triangle $\Delta ABC$ is $A=49.7°$ .

The law of Sines for triangle $\Delta ABC$ ,with sides $a,b,andc$ and opposite angles with measures $A,B,andC$ is

  $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Use the law of Sines for $\Delta ABC$ with sides $a=200\&b$ and the opposite angle $B=44.3°\&A=49.7°$ to find the side $b$ as follows:

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin B}{b}\\ \frac{\sin 49.7°}{200}=\frac{\sin 44.3°}{b}\\ b=\frac{\sin 44.3°\times 200}{\sin 49.7°}\\ b=\frac{200\times 0.6984}{0.7627}\\ b=\frac{139.68}{0.7627}\\ b\approx 183.1388\end{array}$

Hence, the height of the tower is $183.1388feet$ .