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A baseball leaves a bat at an angle of \(\theta\) with reference to the horizontal. The initial velocity is \(v_0 = 91\) feet per second. The baseball is caught 186 feet away. Find one value of \(\theta\) if the range of the baseball is given by the equation \(r = 132v_0^2\sin(2\theta)\). Express your answer in degrees or radians, rounded to the nearest hundredth.

Answer :

Final answer:

To find the angle θ for a range of 186 feet with an initial velocity of 91 feet per second, we substitute the values into the range equation and solve for θ. The calculation involves taking the arcsine and dividing by 2 to account for sin(2θ), and the final answer is rounded to the nearest hundredth in degrees or radians.

Explanation:

To find the value of the angle θ at which a baseball must be hit to travel 186 feet away, we use the given range formula r = 132v02sin(2θ). The initial velocity (v0) is given as 91 feet per second, and the range (r) is 186 feet.

Substituting r and v0 into the range formula, we get:

186 = 132(91)2sin(2θ)

Simplifying, we can solve for sin(2θ) and then for θ:

sin(2θ) = 186 / [132(91)2]

2θ = arcsin(186 / [132(91)2])

θ = 0.5 * arcsin(186 / [132(91)2])

Calculating this will give us one value of θ (since sin(2θ) will have two solutions for θ in the range [0, π/2] or [0, 90°]), but the question asks for only one value.

Rounding to the nearest hundredth will provide the final answer in either degrees or radians, as requested.

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