The automobile assembly plant you manage has a Cobb-Douglas production function

Question

The automobile assembly plant you manage has a Cobb-Douglas production function given by

P = 10x0.2y0.8

where P is the number of automobiles it produces per year, x is the number of employees, and y is the daily operating budget (in dollars). Assume that you maintain a constant work force of 140 workers and wish to increase production in order to meet a demand that is increasing by 50 automobiles per year. The current demand is 1300 automobiles per year. How fast should your daily operating budget be increasing? HINT [See Example 4.] (Round your answer to the nearest cent.)
$  per year

Explanation / Answer

given P = 10x0.2y0.8

work force of 140 workers, current demand is 1300 automobiles per year =>x=140,P=1300

=>1300=10*1400.2*y0.8

=>y=(1300/(10*1400.2))1/0.8

=>y =127.6

constant work force=> x is constant , production is increasing by 50 automobiles per year=>dP/dt =50

P = 10x0.2y0.8

differentiate with respect to t on both sides

=>(dP/dt) =10x0.2*0.8y0.8-1 *(dy/dt)

=>50 =10*1400.2*0.8*127.60.8-1 *(dy/dt)

=>(dy/dt) =6.13527

daily operating budget be increasing at 6.14 dollars per year

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