Chapter 6.1, Problem 79PE

Pipe for a water line must be installed from a main water line at point A to a building on Hontoon Island State Park at point B as shown in the figure. The cost to install water pipe over land is $ $10$ per foot and the cost to install pipe under water is $ $20$ per foot

a. Write an expression in terms of $6$ to represent the total cost c (in dollars) to lay pipe from point A to point B.

b. Use the TABLE function on a calculator to find the cost for $6=20°,25°,30°,35°,$ and $40°$ . Round to 1 decimal place.

c. Which angle from part (b) yields the least cost?

d. Using calculus, we can show that the angle needed to minimize the total cost is a solution to the equation $4000\sec \theta \tan \theta -2000{\sec }^2=0$ . Solve the equation for $6$ , where $0°<\theta <90°.$