Chapter 1, Problem 10PS

a.

To determine

Find the total time $T$ as a function of the distance $x$ .

Answer to Problem 10PS

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

Explanation of Solution

Calculation:

Consider the following figure

  

The speed at which the boat is rowed is $2$ miles per hour.

The distance, travelled by rowing is

  $d_t=\sqrt{2^2+x^2}$

Since, the speed of rowing is $2$ miles per hour hence the time taken to cover a distance of

  $\begin{array}{l}{d}_t=\sqrt{2^2+x^2}\\ t_t=\frac{\sqrt{2^2+x^2}}{2}\end{array}$

Now, the distance travelled by walking is

  $d_w=\sqrt{1^2+{(3-x)}^2}$

Since, the speed of waalking is $4$ miles per hour hence the time taken to cover a distance of

  $d_w=\sqrt{1^2+{(3-x)}^2}$ miles is

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

The total time taken to reach the point $Q$ is

  $\begin{array}{l}T=t_t+t_w\\ =\frac{\sqrt{2^2+x^2}}{2}+\frac{\sqrt{1^2+{(3-x)}^2}}{4}\\ =\frac{2\sqrt{4+x^2}+\sqrt{1+9+x^2-6x}}{4}\end{array}$

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

Hence, the total time $T$ as a function of the distance $x$ is $\frac{2\sqrt{4+x^2}+\sqrt{x^2-6x+10}}{4}$

b.

To determine

Determine the domain of the function.

Answer to Problem 10PS

The domain of the function is all real numbers.

Explanation of Solution

Calculation:

Consider the following figure

  

The function $T=\frac{2\sqrt{4+x^2}+\sqrt{x^2-6x+10}}{4}$ is defined for all values of $x$ .

Hence, the domain of the function is all real numbers.

c.

To determine

Use a graphing utility to graph the function.

Answer to Problem 10PS

  

Explanation of Solution

Calculation:

To graph the function enter the function in Ti-83 at $Y=$ as follows

  

Choose the appropriate WINDOW as follows

  

Press GRAPH button to get the graph as follows

  

Hence, the result is shown in graph.

d.

To determine

Find the value of $x$ that minimizes $T$ .

Answer to Problem 10PS

  $T=1.67$ , $x=1$

Explanation of Solution

Calculation:

Consider the following figure

Press the TRACE button to get the point where $T$ has minimum value as follows

  

Hence, the minimum value of the function is $T=1.67$ approximately at $x=1$ .

e.

To determine

Write a brief paragraph interpreting these values.

Answer to Problem 10PS

  $T=1.67$ hours.

Explanation of Solution

Calculation:

Hence, the minimum total time taken to reach the point $Q$ is $T=1.67$ . This implies that the minimum total time is taken when the distance, $x$ , is $1$ mile down the coast and it is $1.67$ hours.