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 46°. the height of the leaning tower of pisa is about 40.1 m 55.5 m 62.6 m 72.0 m

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In triangle \(ABC\), we know \(B = 50^{\circ}\), \(C=46^{\circ}\), and \(AB = 52.1\) meters. First, we find angle \(A\). Since the sum of angles in a triangle is \(180^{\circ}\), \(A = 180^{\circ}-(50^{\circ}+46^{\circ})=84^{\circ}\). We want to find \(b\) (the side opposite angle \(B\)). Using the Law of Sines formula \(\frac{b}{\sin B}=\frac{AB}{\sin C}\).

Step2: Substitute the values

Substitute \(B = 50^{\circ}\), \(AB = 52.1\), and \(C = 46^{\circ}\) into the formula \(\frac{b}{\sin50^{\circ}}=\frac{52.1}{\sin46^{\circ}}\). Then \(b=\frac{52.1\times\sin50^{\circ}}{\sin46^{\circ}}\).
We know that \(\sin50^{\circ}\approx0.7660\) and \(\sin46^{\circ}\approx0.7193\).

$$b=\frac{52.1\times0.7660}{0.7193}=\frac{39.90}{0.7193}\approx55.5$$

Answer:

\(55.5\) meters