Chapter 3.6, Problem 37E

a.

To determine

To find: the function to evaluate the cost of laying the cable.

Answer to Problem 37E

  $f\left(x\right)=5000\sqrt{x^2+4}+3500\left(10-x\right)$

Explanation of Solution

Given information:

Let $x$ be the distance in kilometers from point B to M. The cost of cable laying underground is $3500 per kilometer and cost of cable laying under water is $5000 per kilometer

  $\begin{array}{l}BM=x\\ CM=10-x\\ AB=2\\ BC=10\end{array}$

Calculation:

The distance underwater is evaluated by using Pythagorean identity as follows:

  $\begin{array}{c}AM=\sqrt{x^2+2^2}\\ =\sqrt{x^2+4}\end{array}$

Thus, the cost of cable laying underwater is $5000\sqrt{x^2+4}$ .

The underground distance for which the cable is laying = $\left(10-x\right)\text{km}$ . Thus, the cost of cable laying underground is $3500\left(10-x\right)$ .

Thus, the function representing the total cost for laying the cable i.e. underground as well as underwater is $f\left(x\right)=5000\sqrt{x^2+4}+3500\left(10-x\right)$ .

b.

To determine

To find: the value of $x$ in order to minimize the cost.

Answer to Problem 37E

  $1.96\text{ km}$

Explanation of Solution

Given information:

Let $x$ be the distance in kilometers.

The maximum cost function evaluated in part a. is shown below.

  $f\left(x\right)=5000\sqrt{x^2+4}+3500\left(10-x\right)$ .

Calculation:

Use a graphing calculator to determine the minimum point as shown below.

  

Since the minimum cost is $42141.428 for 1.96 km. Thus, the required value of $x$ is 1.96.