Chapter 9.5, Problem 72E

a.

To determine

To find: The equation that models the using the function $h=-16t^2+v_{0}t+s_0$ .

Answer to Problem 72E

The required equation is $h=-16t^2+45t+2.5$ .

Explanation of Solution

Given information:

Initial height = 2.5 feet

Initial vertical velocity = 45 feet per second

The given initial height is 2.5 feet that is, $s_0=2.5$ feet

And, the initial vertical velocity is 45 feet per second that is, $v_0=45$ feet per second

The given equation is $h=-16t^2+v_{0}t+s_0$ .

Substitute 45 for $v_0$ and 2.5 for $s_0$ in the above equation.

  $h=-16t^2+45t+2.5$

Hence, the required equation is $h=-16t^2+45t+2.5$ .

b.

To determine

To calculate: The amount of time that the football is in the air.

Answer to Problem 72E

The required amount of time that the football is in the air is 2.74 seconds.

Explanation of Solution

Given information:

Initial height = 2.5 feet

Initial vertical velocity = 45 feet per second

The height of the football = 5.5 feet

The given initial height is 2.5 feet that is, $s_0=2.5$ feet

The initial vertical velocity is 45 feet per second that is, $v_0=45$ feet per second

And, the height of the football is 5.5 feet that is, $h=5.5$ feet

The given equation is $h=-16t^2+v_{0}t+s_0$ .

Substitute 45 for $v_0$ , 5.5 for h and 2.5 for $s_0$ in the above equation.

  $\begin{array}{c}5.5=-16t^2+45t+2.5\\ 16t^2-45t-2.5+5.5=0\\ 16t^2-45t+3=0\end{array}$

Simplify the above equation using the Quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

  $\begin{array}{l}t=\frac{-\left(-45\right)\pm \sqrt{{\left(-45\right)}^2-4\left(16\right)\left(3\right)}}{2\left(16\right)}\\ t=\frac{45\pm \sqrt{1833}}{32}\end{array}$

Solve the above expression for t .

  $\begin{array}{l}t=\frac{45+\sqrt{1833}}{32}\\ t=\frac{45+42.81}{32}\\ t\approx 2.74\end{array}$

And, $\begin{array}{l}t=\frac{45-\sqrt{1833}}{32}\\ t=\frac{45-42.81}{32}\\ t=\frac{2.19}{32}\\ t\approx 0.07\end{array}$

The value t = 0.07 second is the first time the ball reaches 5.5 feet.

So, the football is in the air for 2.74 seconds.

Hence, the required amount of time that the football is in the air is 2.74 seconds.