Suppose you work for a construction company that has a contract to build a new bridge across Scorpion Gulch, downstream from the present bridge. You collect information from various sources in order to decide just where to build the bridge. From surveyors, you find that the width of the gulch is approximately w=10(x^2-8x+22) ft.; where x is the number of miles downstream from the old bridge. The depth of the water is d=20x+10 ft.

a. The structural department specifies that the bridge can be built in water as much as 130 feet deep, but that it can be no more than 310 feet long. The city traffic department specifies that the bridge must be at least one mile downstream from the present bridge. What is the domain of x?

b. What are the shortest and longest lengths the bridge could be?

c. The cost of building the bridge is proportional to the product of the length of the bridge and the depth of the water. Where should the bridge be built to minimize the cost?