Math 21a Math21a, Fall 2001
Course Head: Prof. Clifford H. Taubes

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Answers to selected problems in the Handout on 3-variable Lagrange multipliers:

5) The maximum, 14, occurs where x = 7 and y = 0. The minumum, -14, occurs where x = -7 and y = 0.
6) The problem is to minimize f(x,y) = 60 x + 100 y subject to the constraint 500 x0.4 y 0.8 = 104. The minimum, 2400 (5/6)4/5 occurs at x = 20(5/6)(5/6)2/3 and y = 24(5/6)(5/6)-1/3.
7) Denote the radius of the cylinder by r and the height by h. You are being asked to minimize (r2 + 2 h r) subject to the constraint r2 h = 50. The minimum occurs where r = (50/)1/3 and h = (50/)1/3.
8) The point has coordinates (101)-1/2 (8, 27, 4). One way to find this point is to realize that the gradient of x2/4 + y2/9 + z2/4 is normal to the plane 2x + 3y + z at the points with minimum and maximum distance.
9) Suppose that corners of the box have coordinates (x, y, z), where x, y and z are positive. These corners will lie on the ellipse (otherwise, the box could be made bigger). Thus, you are asked to maximize the volume, 8xyz, subject to the constraint x2/4 + y2/9 + z2/4 = 1. The maximum occurs where x = 2 3-1/2, y = 31/2, z = 2 3-1/2.
10) An open-top rectangular box of side length x, y, and z (height) has volume xyz and surface area that is equal to xy + 2(xz + yz) = 36. The maximum volume is z=3 1/2 for x = y = 2 31/2 and z =31/2. Meanwhile, an open-top cylindrical box with radius r and height z has volume r2h and surface area r2 + 2rh = 36. The maximum volume here is 24 (3/)1/2 which occurs, when r = h = (12/)1/2. Thus, the cylindrical box has 8 () -1/2 times as much volume as the rectangular one.
11) You are being asked to minimize the function 80x + 25y + 15z subject to the constraint that 300x2/5y1/2z1/10 = 12,000. The minimum occurs for x = 10 (768)1/10, y = 40 (768)1/10 and z = 40 (786)1/10/3.
12a) You are being asked to minimuze the function 35x + 16y subject to 500x7/10 y1/2 = 40,000. The minimum occurs at x = 32, y = 50.
b) You are being asked to maximize 500x7/10y1/2 subject to 35x + 16y = 4800. The maximum occurs at x = 80, y = 125.
13a) At ten months, x = 100, y = 125 so the money being spent is 100x + 120y = 25,000 and the production is 300x1/2 y1/3 = 15,000.
13b) You are asked to evaluate P at x = 100 and y = 125. The answer is 75.

13c)
You are asked to evaluate P at x=100 and y=125. The answer is 40.
13d) You are asked to maximize 300x1/2y1/3 subject to 100x + 120y = 25,000. The maximum occurs at x = 150, y = 250/3 and equals 7500 25/6 31/6.
13e) You are asked to evaluate P at x = 150, y = 250/3. The answer is 25 25/6 31/6.

Last update, 09/14/2001