The rectangular plot in Figure 27 has size 100 m × 200 m. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying pipe along the side of the plot is $45/m and the cost through the plot is $80/m (since it is underground).

(a) Let f(x) be the total cost, where x is the distance from P to B. Determine f(x), but note that f is discontinuous at x = 0 (when x = 0, the cost of the entire pipe is $45/m). (b) What is the most economical way to lay the pipe? What if the cost along the sides is $65/m? B 100 A X 200 P 200 x C