Question

a baseball leaves a bat at an angle of $\theta$ in reference to the horizontal. the initial velocity is $v_0 = 81$ feet per second. the baseball is caught 166 feet away. find one value of $\theta$ if the range of the baseball is given by the equation $r=\frac{1}{32}v_0^{2}sin(2\theta)$. express your answer in degrees or radians rounded to the nearest hundredth.

Explanation:

Step1: Substitute given values into formula

We are given $r = 166$, $v_0=81$. Substitute into $r=\frac{1}{32}v_0^{2}\sin(2\theta)$. So, $166=\frac{1}{32}\times81^{2}\sin(2\theta)$.

Step2: Simplify the right - hand side

First, calculate $\frac{1}{32}\times81^{2}=\frac{6561}{32}$. The equation becomes $166 = \frac{6561}{32}\sin(2\theta)$.

Step3: Solve for $\sin(2\theta)$

Multiply both sides by $\frac{32}{6561}$: $\sin(2\theta)=\frac{166\times32}{6561}=\frac{5312}{6561}\approx0.81$.

Step4: Find $2\theta$

Using the inverse - sine function, $2\theta=\sin^{- 1}(0.81)$. In radians, $2\theta\approx0.95$ (rounded to the nearest hundredth). In degrees, $2\theta=\sin^{-1}(0.81)\times\frac{180^{\circ}}{\pi}\approx54.46^{\circ}$.

Step5: Solve for $\theta$

Divide by 2. In radians, $\theta=\frac{0.95}{2}=0.48$ (rounded to the nearest hundredth). In degrees, $\theta=\frac{54.46^{\circ}}{2}=27.23^{\circ}$.

Answer:

In degrees: $27.23^{\circ}$; In radians: $0.48$