Question

anik built two snowball catapults. the catapults are 14.5 m apart, as shown. from catapult a, there is a 96° angle between catapult b and the target. from catapult b, there is a 58° angle between catapult a and the target. how far should catapult b throw the snowball to reach the target? do not round during your calculations. round your final answer to the nearest tenth of a 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 target be \(C\). So, \(C = 180^\circ - 96^\circ - 58^\circ = 26^\circ\).

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 catapult B to the target be \(b\), the distance between A and B be \(c = 14.5\) m, angle at A be \(A = 96^\circ\), angle at target be \(C = 26^\circ\). So, \(\frac{b}{\sin 96^\circ}=\frac{14.5}{\sin 26^\circ}\).

Step3: Solve for \(b\)

\(b=\frac{14.5\times\sin 96^\circ}{\sin 26^\circ}\). Calculate \(\sin 96^\circ\approx0.9945\) and \(\sin 26^\circ\approx0.4384\). Then \(b=\frac{14.5\times0.9945}{0.4384}\approx\frac{14.42025}{0.4384}\approx32.9\).

Answer:

\(32.9\)