Two baseball players are warming up by throwing balls to the catcher. The two players are located 10 yards apart, and the angle from each player to the catcher is 55^(@) and 71^(@), respectively. What is the distance from the farthest baseball player to the catcher? [5.02] A. 10.125 yards B. 11.687 yards C. 14.501 yards D. 14.790 yards

Question

Two baseball players are warming up by throwing balls to the catcher. The two players are located 10 yards apart, and the angle from each player to the catcher is  55^(@) and  71^(@), respectively. What is the distance from the farthest baseball player to the catcher?  [5.02] A. 10.125 yards B. 11.687 yards C. 14.501 yards D. 14.790 yards

Bytelearn · Accepted Answer

Understand and Visualize: Understand the problem and visualize the scenario. We have a triangle where the distance between the two baseball players is one side, and the distances from each player to the catcher form the other two sides. The angles given are from each player to the catcher. We need to find the distance from the farthest player to the catcher, which is opposite the largest angle.Use Law of Sines: Use the law of sines to set up the equation. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Let's denote the distance from the farthest player to the catcher as $ d $, the distance between the players as $ a = 10 $ yards, and the angles as $ \alpha = 55^\circ $ and $ \beta = 71^\circ $. The angle opposite the side we want to find is $ \gamma = 180^\circ - \alpha - \beta $.Calculate Opposite Angle: Calculate the angle opposite the side we want to find. $ \gamma = 180^\circ - 55^\circ - 71^\circ $ $ \gamma = 180^\circ - 126^\circ $ $ \gamma = 54^\circ $Apply Law of Sines: Apply the law of sines to find the distance $ d $. Using the law of sines: $ \frac{a}{\sin(\gamma)} = \frac{d}{\sin(\beta)} $ $ \frac{10}{\sin(54^\circ)} = \frac{d}{\sin(71^\circ)} $Solve for Distance: Solve for $ d $. $ d = \frac{10 \cdot \sin(71^\circ)}{\sin(54^\circ)} $ Now we calculate the sine values and solve for $ d $. $ d \approx \frac{10 \cdot 0.9455}{0.8090} $ $ d \approx \frac{9.455}{0.8090} $ $ d \approx 11.687 $ yards

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