Chapter 9, Problem 93RE

To determine

Find the maximum height of the football and also draw the graph of path of the ball.

Answer to Problem 93RE

The maximum height of the football is $23.8feet$ .

Explanation of Solution

Given: It is given in the question that the quarterback of a football team releases a pass at a height $7$ feet above the playing field, and the football is caught at a height of $4$ feet, $30$ yards directly downfield. The pass is released at an angle of ${35}^{\circ }$ with the horizontal. The parametric equations for the path of the football are given by $x=0.82v_{{}^{\circ }}\mathrm{t}andy=7+0.57v_{{}^{\circ }}t-t^2$ , where $v_{{}^{\circ }}$ is the speed of the football (in feet per second) when it is released.

Concept Used:

In this, use the concept of projectile motion and understand the question and do the answer. Use the y coordinate to determine the maximum height.

Calculation:

Here, the two equations are $x=44.36tandy=7+30.83t-16t^2$

Use the y coordinate equation, to find the maximum height:

  $\begin{array}{l}y(t)=7+30.83t-16t^2\\ y^{\text{'}}(t)=30.83-32t\end{array}$

For maximum height , $y^{\text{'}}(t)=0$

  $\begin{array}{l}{y}^{\text{'}}(t)=0\\ 30.83-32t=0\\ 32t=30.89\\ t=0.9697\end{array}$

Now put the value of t in equation of y,

  $\begin{array}{l}y(0.9697)=7+30.83(0.9697)-16{(}^{0.9697}\\ y(0.9697)=7+29.835-12.96\\ y(0.9697)=23.8\end{array}$

The graph for the equation are given below:

  

Conclusion:

The maximum height is $23.8feet$ .