Chapter 4, Problem 16STP

(a)

To determine

To plot: The graph of the function $h\left(t\right)=-16t^2+100t$ on a coordinate grid.

Answer to Problem 16STP

The graph of the function $h\left(t\right)=-16t^2+100t$ is shown in figure (1).

Explanation of Solution

Given:

The function is $h\left(t\right)=-16t^2+100t$ .

Calculation:

Plot the graph of the $h\left(t\right)=-16t^2+100t$ as follows.

  

Figure (1)

(b)

To determine

To find: The maximum height reached by the football.

Answer to Problem 16STP

The maximum height is $156.25\text{feet}$ .

Explanation of Solution

Given:

The function is $h\left(t\right)=-16t^2+100t$ .

Calculation:

To find the maximum height first find the value of $t$ by solving the equation $\frac{dh}{dt}=0$ .

Differentiate the equation with respect to $t$ .

  $\begin{array}{c}\frac{dh}{dt}=-16\left(2t\right)+100\\ =-32t+100\end{array}$

Equate $\frac{dh}{dt}=0$ .

  $\begin{array}{c}-32t+100=0\\ 32t=100\\ t=\frac{100}{32}\\ =3.125\end{array}$

Substitute $t=3.125$ in the equation $h\left(t\right)=-16t^2+100t$ to find the maximum height.

  $\begin{array}{c}h\left(t\right)=-16{\left(3.125\right)}^2+100\left(3.125\right)\\ =-156.25+312.5\\ =156.25\text{feet}\end{array}$

Therefore, the maximum height is $156.25\text{feet}$ .

(c)

To determine

To find: The time taken by the football in the air before it hits the ground.

Answer to Problem 16STP

The time is $3.125$ seconds.

Explanation of Solution

Given:

The function is $h\left(t\right)=-16t^2+100t$ .

Calculation:

Substitute $156.25\text{feet}$ in the equation $h\left(t\right)=-16t^2+100t$ to find the value of $t$ .

  $\begin{array}{c}156.25=-16t^2+100t\\ 16t^2-100t+156.25=0\end{array}$

Solve the quadratic equation.

  $\begin{array}{c}t=\frac{100\pm \sqrt{{\left(-100\right)}^2-4\left(16\right)\left(156.25\right)}}{2\left(16\right)}\\ =\frac{100\pm 0}{32}\\ =\frac{100}{32}\\ =3.125\end{array}$

Therefore, the time is $3.125$ seconds.