Chapter 1, Problem 10PS

(a)

To determine

The total time $T$ the trip as a function of $x$ .

Answer to Problem 10PS

The total time of the trip as the function of $x$ is $\frac{2\sqrt{2^2+x^2}+\sqrt{x^2-6x+10}}{4}$ .

Explanation of Solution

Given:

Consider the layout of the given diagram is shown in Figure 1

  

Figure 1

Calculation:

The speed at which the boat travelled per hour is $2\text{mile}$ per hour and the distance travelled by the boat is,

  $d_t=\sqrt{2^2+x^2}$

Since the boat is moving at the speed of $2\text{mile}$ per hour the time taken to cover the distance is,

  $t_t=\frac{\sqrt{2^2+x^2}}{2}$

The distance travelled by walking is,

  $d_w=\sqrt{1^2+{\left(3-x\right)}^2}$

The speed of walking is $4\text{mile}$ per hour, so the time taken to cover the distance by walking is,

  $t_w=\frac{\sqrt{1^2+\left(3-x^2\right)}}{4}$

Then, the total time taken to reach the point Q is,

  $T=t_t+t_w$

Then,

  $\begin{array}{c}T=\frac{\sqrt{2^2+x^2}}{2}\frac{\sqrt{1^2+\left(3-x^2\right)}}{4}\\ =\frac{2\sqrt{2^2+x^2}+\sqrt{x^2-6x+10}}{4}\end{array}$

(b)

To determine

The domain of the function.

Answer to Problem 10PS

The domain of the function is all the real numbe.

Explanation of Solution

Calculation:

The total time taken of the trip is,

  $T=\frac{2\sqrt{2^2+x^2}+\sqrt{x^2-6x+10}}{4}$

The value of $T$ is defined for all the values taken by $x$ therefore the domain of the function is all the real number.

(c)

To determine

The graph for the function.

Answer to Problem 10PS

The graph is shown in Figure 4

Explanation of Solution

Calculation:

To graph the function enter the function in Ti-83=as follows,

  

Figure 2

The second step is to choose the appropriate window,

  

Figure 3

The third step is to press the graph button as,

  

Figure 4

(d)

To determine

The values of the variable $x$ that minimizes $T$ .

Answer to Problem 10PS

The value of the variable is $x$ .

Explanation of Solution

Calculation:

On the calculator after obtaining the graph press the trace button to button to obtain the point where the $T$ has the maximum value,

The required graph is,

  

Figure 5

The maximum value of the function is $T$ equal to $1.67$ that is at $x=1$ .

(e)

To determine

The brief paragraph to interrupt the values.

Answer to Problem 10PS

The minimum total time taken for the distance of 1 mile is $1.67\text{hours}$

Explanation of Solution

The minimum time $T$ taken to reach the destinationis $T=1.67$ . This suggest that the minimum total time taken for the distance $x$ is 1 mile down the coast that is $1.67\text{hours}$ .