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Your automobile assembly plant has a Cobb-Douglas production function given by
q = x0.4y0.6,
where q is the number of automobiles it produces per year, x is the number of employees, and y is the daily operating budget (in dollars). Annual operating costs amount to an average of $26,000 per employee plus the operating budget of $365y. Assume that you wish to produce 1,000 automobiles per year at a minimum cost. How many employees should you hire? (Round your answer to the nearest employee.)

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MissPhiladelphia
These are the important equations:

(1)  q = 0.4x0.6y
(2)  r = 26000x + 365y   (r is the annual operating costs)

Since q is given (q=1000), we can express y in terms of x:

1000 = 0.4x0.6y
y = 1000/[(0.4x)(0.6)]
y = 12500/3x   ----> we substitute this to equation 2

r = 26000x + 365(12500/3x)
r = 26000x + 1520833.33x^-1
r should be minimized. In calculus, we differentiate this and equate to zero. Then we can determine x, which is the number of employees.

dr/dx = 26000x - 1520833.33x^-2 = 0
1520833.33/x^2 = 26000
26000x^2 = 1520833.33
x^2 = 58.4935
x = 7.6
x=8

Thus, you should employ 8 employees to produce 1,000 automobiles a year with a minimum annual operating cost.