Chapter 1.5, Problem 67E

(a)

To determine

To find:The function that represent the situation of the given equation.

Answer to Problem 67E

The function that represent the situation of the given equation is $s=-16t^2+64t+6$ .

Explanation of Solution

Given information:

The given equationis $s=-16t^2+v_{0}t+s_0$ .

The height is $s_0=6\text{feet}$ .

The initial velocity is $v_0=64\text{feet/s}$ .

Calculation:

Using the position $s=-16t^2+v_{0}t+s_0$ .

The function is.

  $s=-16t^2+64t+6$

Therefore, the function that represent the situation of the given equation is $s=-16t^2+64t+6$ .

(b)

To determine

To graph: For the given function.

Explanation of Solution

Given information:

The given function is $s=-16t^2+64t+6$ .

Graph:

The graph for the given function is shown in figure (1).

  

Figure (1)

Interpretation: Graph for the function $s=-16t^2+64t+6$ is shown in figure (1).

(c)

To determine

To find: The average rate of change of the function from $t_1$ to $t_2$ .

Answer to Problem 67E

The average rate of change of the function from $t_1$ to $t_2$ is $16$ .

Explanation of Solution

Given information:

The given equation is $s=-16t^2+64t+6$ .

The values are $t_1=0$ and $t_2=3$ .

Calculation:

Average rate of change of the function from $t_1$ to $t_2$ .

  $\frac{f\left(t_2\right)-f\left(t_1\right)}{t_2-t_1}$

Calculate the values at $t_1=0$ and $t_2=3$ .

  $\begin{array}{c}f\left(0\right)=-16{\left(0\right)}^2+64\left(0\right)+6\\ =6\\ f\left(3\right)=-16{\left(3\right)}^2+64\left(3\right)+6\\ =54\end{array}$

Calculate average rate of change of the function from $t_1$ to $t_2$ .

  $\begin{array}{c}=\frac{f\left(3\right)-f\left(0\right)}{3-0}\\ =\frac{54-6}{3}\\ =\frac{48}{3}\\ =16\end{array}$

Therefore, the average rate of change of the function from $t_1$ to $t_2$ is $16$ .

(d)

To determine

To find: The behavior of slope of the secant line through $t_1$ to $t_2$ .

Answer to Problem 67E

The slope of the secant line through $t_1$ to $t_2$ is positive.

Explanation of Solution

Given information:

The given equation is $s=-16t^2+64t+6$ .

The values are $t_1=0$ and $t_2=3$ .

Calculation:

The average rate of change between any two points $\left(t_1,f\left(t_1\right)\right)$ and $\left(t_2,f\left(t_2\right)\right)$ is the slope of the line secant line is between two points $\left(0,6\right)$ and $\left(3,54\right)$ .

Therefore, the slope of the secant line through $t_1$ to $t_2$ is positive.

(e)

To determine

To find: The slope of the secant line through $t_1$ to $t_2$ .

Answer to Problem 67E

The slope of the secant line through $t_1$ to $t_2$ is $f\left(t\right)=16t+6$ .

Explanation of Solution

Given information:

The given equation is $s=-16t^2+64t+6$ .

The values are $t_1=0$ and $t_2=3$ .

Calculation:

Using the above value.

  $\frac{f\left(t_2\right)-f\left(t_1\right)}{t_2-t_1}=16$

The equation of secant line is.

  $\begin{array}{c}\left(f\left(t\right)-f\left(0\right)\right)=\frac{f\left(t_2\right)-f\left(t_1\right)}{t_2-t_1}\left(t-0\right)\\ \left(f\left(t\right)-6\right)=16t\\ f\left(t\right)=16t+6\end{array}$

Therefore, the slope of the secant line through $t_1$ to $t_2$ is $f\left(t\right)=16t+6$ .

(f)

To determine

To graph: For the secant line.

Explanation of Solution

Given information:

The secant line is $f\left(t\right)=16t+6$ .

Graph:

The graph for the secant line is shown in figure (1).

  

Figure (1)

Interpretation: Graph for the secant line $f\left(t\right)=16t+6$ is shown in figure (1).