Chapter 1.5, Problem 49E

Solution

a.

To Graph: The parametric equations which represent the position of a fly ball and the fence at the boundary respectively for $0\le t\le 6$ :

  $\begin{array}{ccc}\begin{array}{l}x=110t\cos \left({60}^{°}\right)\\ y=110t\sin \left({60}^{°}\right)-16t^2\end{array}& \text{and}& \begin{array}{l}x=325\\ y=5t\end{array}\end{array}$

To State: Whether the fly ball clears the fence.

The fly ball does not clear the fence.

Given: The parametric equations which represent the position of a fly ball and the fence at the boundary respectively:

  $\begin{array}{ccc}\begin{array}{l}x=110t\cos \left({60}^{°}\right)\\ y=110t\sin \left({60}^{°}\right)-16t^2\end{array}& \text{and}& \begin{array}{l}x=325\\ y=5t\end{array}\end{array}$

The allowed values $0\le t\le 6$ .

Concepts Used:

The function call $\mathrm{Curve}(110t\cos (c^{°}),110t\sin (c^{°})-16t^{(2)},t,0,6)$ can be used to graph the trajectory of the fly ball when it is launched at an initial angle of $c^{°}$ .

The function call $\mathrm{Curve}(325,5t,t,0,6)$ can be used to graph the fence.

For the fly ball to clear the fence, the graph of trajectory of ball must pass above the fence without touching it.

Graph:

  

Interpretation:

The trajectory of the fly ball has not passed completely above the fence. It hits the fence near its bottom. So, the fly ball does not clear the fence.

b.

To Graph: The parametric equations given in the previous part for the angle ${30}^{°}$ instead of ${60}^{°}$ .

To State: Whether the fly ball clears the fence.

The fly ball does not clear the fence.

Given: The parametric equations which represent the position of a fly ball and the fence at the boundary respectively:

  $\begin{array}{ccc}\begin{array}{l}x=110t\cos \left({30}^{°}\right)\\ y=110t\sin \left({30}^{°}\right)-16t^2\end{array}& \text{and}& \begin{array}{l}x=325\\ y=5t\end{array}\end{array}$

The allowed values $0\le t\le 6$ .

Concepts Used:

The function call $\mathrm{Curve}(110t\cos (c^{°}),110t\sin (c^{°})-16t^{(2)},t,0,6)$ can be used to graph the trajectory of the fly ball when it is launched at an initial angle of $c^{°}$ .

The function call $\mathrm{Curve}(325,5t,t,0,6)$ can be used to graph the fence.

For the fly ball to clear the fence, the graph of trajectory of ball must pass above the fence without touching it.

Graph:

  

Interpretation:

The trajectory of the fly ball has not passed completely above the fence. It hits the fence near its bottom. So, the fly ball does not clear the fence.

c.

To Determine: The optimal angle for hitting the ball.

To State: Whether the fly ball clears the fence.

The optimal angle is ${45}^{°}$ . The fly ball clears the fence.

Given: The parametric equations which represent the position of a fly ball and the fence at the boundary respectively:

  $\begin{array}{ccc}\begin{array}{l}x=110t\cos \left(c^{°}\right)\\ y=110t\sin \left(c^{°}\right)-16t^2\end{array}& \text{and}& \begin{array}{l}x=325\\ y=5t\end{array}\end{array}$

The allowed values $0\le t\le 6$ .

Concepts Used:

The function call $\mathrm{Curve}(110t\cos (c^{°}),110t\sin (c^{°})-16t^{(2)},t,0,6)$ can be used to graph the trajectory of the fly ball when it is launched at an initial angle of $c^{°}$ .

The function call $\mathrm{Curve}(325,5t,t,0,6)$ can be used to graph the fence.

For the fly ball to clear the fence, the graph of trajectory of ball must pass above the fence without touching it.

Graph:

By the process of hit and trial to plot the trajectory of the ball for various angles of hitting the ball, it was found that maximum distance was covered when the angle of hitting was ${45}^{°}$ . The resulting graph is shown below.

  

Interpretation:

The optimal angle is ${45}^{°}$ . The trajectory of the fly ball has passed completely above the fence. So, the fly ball clears the fence.