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

It turns out that the lowest point in the valley that you found in question #2 (x = 3, y = 4) is owned by an obstreperous farmer who will not allow the county to drill. Fortunately, the county owns a small piece of land in the southwestern-most corner of the county bounded by the line described by 2x + 2y = 6. That is, the parcel of land contains all of the points with x greater than or equal to 0, y greater than or equal to 0, and 2x + 2y less than or equal to 6.

a. At what point within this parcel of land should the county drill its well? (I.e. find the lowest point in this parcel.) Show your work. (Hint: You need to set up a Lagrangian for this one.)

Answer 1

CATE: 2 | тіл. ; _ A = f(x 2 ) = lbs + 4* - Чаа 2 - 1х - subject to : ax+2y = 6 9-о 12 da setting up Lagragean: 4 = f(x) + A [ 5- 2x – 24] -= Isus + Ча* -Ч78 + 2+ — - Чjt L6-24-) - - Чая) – ч — + A(-а) го . – Ч — + — 21-о (0) - - - Ч7+) –ч + + (-2) го — -“х м--- ад - о -(3) 2. 2 - 2 1-2то 2 + 2 = 6 1-+ 1 = 2 — (3) Rear sie Caph.. 5 : -1 - 4 = 4, Rearranging equation : 4x + 4y -4 = 22 А дъ-Чу — = -4х + Ч7 - .. 121- #ч ч 12 (3- у - у 1 3. — 2- Г. - 1 - - +36 - - - Чhin (3)

{ x= 3-4 = 3-1.6 = 1.4 so, at n=14 y=1:6 should the country drill its well-