Question

two construction workers, tyrone and diego, are standing 15 yards apart, looking up at a piano hanging between the two men from a crane. tyrone is staring at the piano with an angle of elevation of 65°, and diego is looking up with an angle of elevation of 77°. what is the distance between tyrone and the piano? round the answer to the nearest tenth.

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\).
\(A = 180^{\circ}-65^{\circ}-77^{\circ}=38^{\circ}\)

Step2: Apply the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, let \(a = 15\) (distance between Tyrone and Diego), \(A = 38^{\circ}\), \(B = 77^{\circ}\), and \(b=x\) (distance between Tyrone and the piano).
\(\frac{15}{\sin38^{\circ}}=\frac{x}{\sin77^{\circ}}\)

Step3: Solve for \(x\)

Cross - multiply: \(x=\frac{15\times\sin77^{\circ}}{\sin38^{\circ}}\)
We know that \(\sin77^{\circ}\approx0.9744\) and \(\sin38^{\circ}\approx0.6157\)
\(x=\frac{15\times0.9744}{0.6157}=\frac{14.616}{0.6157}\approx23.7\)

Answer:

\(23.7\) yards