Question

the leaning tower of pisa in italy leans at an angle of about 84°. the figure below shows 52.1 meters from the base of the tower, the angle of elevation to the top is 50°. find the distance (b) from the base to the top of the tower. the measure of angle c is

Explanation:

Step1: Find angle \( C \)

Use the angle - sum property of a triangle (\( A + B + C=180^{\circ} \)). Given \( A = 84^{\circ}\), \( B = 50^{\circ}\).
\( C=180^{\circ}-(84^{\circ}+50^{\circ})\)
\( C = 46^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here \( a = 52.1\), \( A = 46^{\circ}\), \( B = 50^{\circ}\).
We want to find \( b\), so \(\frac{b}{\sin B}=\frac{a}{\sin C}\)
\( b=\frac{a\sin B}{\sin C}\)
Substitute \( a = 52.1\), \( B = 50^{\circ}\), \( C = 46^{\circ}\)
\( b=\frac{52.1\times\sin(50^{\circ})}{\sin(46^{\circ})}\)
We know that \(\sin(50^{\circ})\approx0.7660\), \(\sin(46^{\circ})\approx0.7193\)
\( b=\frac{52.1\times0.7660}{0.7193}\)
\( b=\frac{39.90}{0.7193}\approx55.5\)

Answer:

The measure of angle \( C\) is \( 46^{\circ}\) and the distance \( b\) from the base to the top of the tower is approximately \( 55.5\) meters.