Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200

Question

Near a buoy, the depth of a lake at the point with coordinates (x, y) is z = 200 + .02x^2 - .001y^3, where, y, and z are measured in meters. A fisherman in a small boat starts at the point (80, 60) and moves toward the buoy, which is located at (0, 0). Is the water under the boat getting deeper or shallower when he departs? Explain. The revenue of a steel foundry is given by R(x, y) = 100x + 50y - x^2 - .3y^2, where x and y are the total feet of I-beams and cables produced by the foundry respectively. The foundry just received a shipment of 50,000 pounds of steel. I-beams require 100 pounds of steel per foot and cables require only 10 pounds per foot. a) How many I-beams and cables should the foundry produce with the recent shipment in order to maximize revenue? b) Find and interpret lambda for the previous problem. c) If the foundry could have ordered 1,000 more pounds of steel for $100, should the foundry have increased their order to 51,000 pounds? Find the dimensions of the box with volume 1000 cm^3 that has minimal surface area. The humidity in a greenhouse is measured by H = 200 - .5x^2 - .4y^2, where humidity is measured in percent and x and y are in meters. If your plant is located at (2, 5). a) What is the direction of greatest decrease in humidity? b) What is the rate of change in the direction 2i + j? The total production of a given product depends on the amount of labor L and the amount of capital investment K. The Cobb-Douglas model for the production function is P = b L^a K^1 - a, such that a and b are positive constants and a

Explanation / Answer

Let us say dimensions of the box be x, y and z.

We know volume of the box will be given by product of its dimensions. Therefore, we can set

xyz=1000

z=1000/(xy)

Surface area of the box would be

A = 2xy+2yz+2xz

A = 2xy+2y (1000/(xy))+2x/(1000/(xy))

A = 2xy+2000/x+2000/y

We find critical points by setting Ax=0 and Ay=0

Ax = 2y-2000/x^2 = 0

Ay = 2x-2000/y^2 = 0

On solving we get,

x=10, y=10

On doing double derivative test we see that at this point surface area is infact minimum.

Therefore, minimum surface area is

A = 2 (10*10) + 2000/10 + 2000/10

A = 200+200+200 =600

Therefore minimum surface area is 600 cm^2

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