Chapter 4.4, Problem 14E

(a)

To determine

To find: The velocity of an object when $t=0$ .

Answer to Problem 14E

96 ft/sec

Explanation of Solution

Given information: The height of an object is given by the function, $s=-16t^2+96t+112$ .

Concept used: The velocity of an object is obtained by differentiating the function of height.

Calculation:

Differentiate the given function with respect to $t$ to obtain the velocity.

  $\begin{array}{c}v\left(t\right)=\frac{ds}{dt}\\ =-32t+96\end{array}$

Substitute $t=0$ into the obtained equation and simplify it to determine the required velocity.

  $\begin{array}{c}v\left(0\right)=-32\left(0\right)+96\\ =96\text{ ft/sec}\end{array}$

(b)

To determine

To find: The maximum height and when that height occurs.

Answer to Problem 14E

The maximum height occurs at $t=3\text{ sec}$ which is equal to 256 ft.

Explanation of Solution

Given information: The height of an object is given by the function, $s=-16t^2+96t+112$ .

Concept used: The maximum height occurs when the velocity of an object will be 0.

Calculation:

Differentiate the given function with respect to $t$ to obtain the velocity.

  $\begin{array}{c}v\left(t\right)=\frac{ds}{dt}\\ =-32t+96\end{array}$

Equate the obtained velocity to 0 and solve for $t$ .

  $\begin{array}{c}v\left(t\right)=0\\ -32t+96=0\\ t=3\end{array}$

Substitute $t=3$ into the given function and simplify it to determine the value of a maximum height.

  $\begin{array}{c}{\left.s\right|}_{t=3}=-16{\left(3\right)}^2+96\left(3\right)+112\\ =256\text{ ft}\end{array}$

(c)

To determine

To find: The velocity of an object when $s=0$ .

Answer to Problem 14E

  $-128\text{ ft/sec}$

Explanation of Solution

Given information: The height of an object is given by the function, $s=-16t^2+96t+112$ .

Concept used:

Calculation:

Substitute $s=0$ into the given function and solve for $t$ .

  $\begin{array}{c}0=-16t^2+96t+112\\ -16\left(t-7\right)\left(t+1\right)=0\\ t=7,-1\end{array}$

The value of $t$ can never be negative. So, the only value of $t$ is 7.

Substitute $t=7$ into the equation, $v\left(t\right)=-32t+96$ , and simplify it to determine the required velocity.

  $\begin{array}{c}v\left(7\right)=-32\left(7\right)+96\\ =-128\text{ ft/sec}\end{array}$