Question

Assume that there is a bowl-shaped valley in the southwestern corner of Lincoln County, Minnesota. Let x = the distance moving east from the southwest corner. Let y = the distance moving north from the southwest corner. Then the altitude (A) in the valley is described by the equation: A = f(x,y) = 1000 + 4x2 - 4xy + 2y2 - 8x - 4y

Highway 14 runs due east/west through the county (horizontal line) at y = 6 (i.e. 6 miles north of the southern border of the county).

a. At what point in the county, driving along Highway 14, will one reach the lowest altitude on the highway? Show your work. (Hint: you are only interested in points where y = 6, so go ahead and substitute y = 6 into f(x,y) above to find the altitude along the highway = f(x,6) = . . . This will be an equation giving altitude along the highway as a function of x alone.)

b. Are you sure you found the lowest point and not the highest point on the highway?

Answer 1