Chapter 2.6, Problem 16E

Laying Phone Cable City A lies on the north bank of a river that is 1 mile wide. You need to run a phone cable from City A to City B, which lies on the opposite bank 5 miles down the river. You will lay L miles of the cable along the north shore of the river, and from the end of that stretch of cable you will lay W miles of cable running under water directly toward City B. (See Figure 2.108.)

Figure 2.108

You will need the following fact about right triangles: A right triangle has two legs, which meet at the right angle, and the hypotenuse, which is the longest side. An ancient and beautiful formula, the Pythagorean theorem, relates the lengths of the three sides:

$\begin{array}{l}\text{Length}\text{of}\text{hypotenuse}\\ \text{=}\sqrt{\text{Length}\text{of}\text{one}{\text{leg}}^{\text{2}}\text{+Length}\text{of}\text{other}{\text{leg}}^{\text{2}}}\end{array}$

a. Find an appropriate right triangle that shows that

$W=\sqrt{1+{(5-L)}^2}$

b. Find a formula for the length of the total phone cable P from City A to City B as a function of L.

c. Make a graph of the total phone cable length P as a function of L, and explain what the graph is showing.

d. What value of L gives the least length for the total phone cable? Draw a picture showing the least-length total phone cable.