Question

curt needs to hit his ball close to the pin, but doesnt know how far away the pin is. curts caddy is 50 meters away from the ball, as shown. from the caddy’s view, there is an 83° angle between curt’s ball and the pin. from the ball, there is a 69° angle between the caddy and the pin. how far is the ball from the pin? do not round during your calculations. round your final answer to the nearest meter.

Explanation:

Step1: Find the third angle of the triangle

The sum of angles in a triangle is \(180^\circ\). Let the angle at the pin be \(A\), at the ball be \(B = 69^\circ\), and at the caddy be \(C = 83^\circ\). Then \(A=180 - 69 - 83=28^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a\), \(b\), \(c\) are the sides opposite angles \(A\), \(B\), \(C\) respectively. Let the distance from the ball to the pin be \(x\) (opposite angle \(C = 83^\circ\)), the distance from the caddy to the ball be \(50\) m (opposite angle \(A = 28^\circ\)). So \(\frac{x}{\sin 83^\circ}=\frac{50}{\sin 28^\circ}\).

Step3: Solve for \(x\)

\(x=\frac{50\times\sin 83^\circ}{\sin 28^\circ}\). Calculate \(\sin 83^\circ\approx0.9925\), \(\sin 28^\circ\approx0.4695\). Then \(x=\frac{50\times0.9925}{0.4695}=\frac{49.625}{0.4695}\approx105.7\). Rounding to the nearest meter gives \(106\).

Answer:

\(106\)