We eliminated y in the given problem to obtain the following optimization problem.

Minimize C = 100x +

160,000 x

with x and y non-negative (y = 10,000/x).

The derivative is

C'(x) = 100 -

160,000 x2

Setting C'(x) = 0 and solving gives

0 = 100 - 160,000 x2

100 = 160,000 x2

100x2 = 160,000

x2 = 1,600

x = 40 laborers.

The coprresponding value of y is given by

y = 10,000/x = 10,000/40 = 250 robots.

Finally, the minimum cost C is obtained by substituting x = 40, y = 250 into the objective function:

C = 100x + 16y = 100(40) + 16(250) = $8,000.

You can now check that the given critical point is indeed a minimum by graphing the objective function

C = 100x +

160,000 x

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