Question: Near its intersection with East-West Highway, Wisconsin Avenue can modeled by the equations

\[y=\left\{ \begin{aligned} & -5.25x\,\,\,\,\,\,\,\,x<0 \\ & -2.78x\,\,\,\,\,\,\,\,x\ge 0 \\ \end{aligned} \right.\]

where (0,0) is the intersection, x and y are distances in meters east and north of the intersection, respectively. Urban planners and trail developers are trying to decide where to relocate the Capital Crescent Trail in the event that the Purple Line takes over the entire

Wisconsin Ave.

tunnel. No matter what happens with the Purple Line, though, the trail will still pass through Little Falls Parkway at (•. 658, —1252), and

Jones Mill Road

at (2471, 1483).

(a) The trail maintenance cost to the west of

Wisconsin Ave.

is $250 per meter, while the corresponding cost to the east of

Wisconsin Ave.

is $800 per meter. Let (x, y) be the point where the trail crosses Wisconsin Ave. Write a formula in terms of (x, y) for the total cost of maintaining the portion of the trail between Little Falls Parkway and Jones Mill Road, for the simple case of line segments connecting these points to Wisconsin Ave. Use the piecewise model of Wisconsin Ave. to rewrite this formula in terms of x alone.

(b) Where would a trail crossing be placed so as to minimize the maintenance costs between Little Falls Parkway and

Jones Mill Road

(your part (a) objective function)?

(c) The cost in thousands of dollars of building a pedestrian/bicycle overpass across

Wisconsin Ave.

is

where h is the distance in meters between the overpass and (0,0). Add this cost to your answer from part (a) to determine the total cost of moving the trail crossing to the point (x, y).

(d) Where would a trail crossing be placed so as to minimize the sum of maintenance and overpass costs (your part (c) objective function)?

(e) What other costs might developers want to consider in their planning? What kinds of functions would you use to model those costs?