Question

Could someone please check my work for this problem.
The automobile assembly plant you manage has a Cobb-Douglas production function given by:

P = 10x^0.5*y^0.5
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 130 workers and wish to increase production in order to meet a demand that is increasing by 50 automobiles per year. The current demand is 1000 automobiles per year. How fast should your daily operating budget be increasing?

Answer 1

Typing out my work right now, give me just a second!!

Answer 2

Typing out my work right now, give me just a second!!

Answer 3

1000 = 5x^-0.5y^0.5 * dx/dt + 0.5y^-.510x^.5 * dy/dt

= 5x^-.5*y^.5 * dp/dt + 5y^-.5*x^.5 * dy/dt
=5(130)^-.5(77.44)^.5(50) + 5(77.44)^-.5(130^.5 * dy/dt
=5(.09)(440) + 5(.11)(11.4) * dy/dt
=192.95 + 6.48 * dy/dt
=199.43*dy/dt
1000/199.43 = dy/dt
dy/dt = 5.01

Answer 4

I got y from this:

P = 10x^0.5y^0.5
1000 = 10(130)^.5*y^.5
1000 = 10(11.4)y^.5
1000 = 114*y^.5
1000/114 =y^.5
8.8 = y^.5
(8.8)^2 = (y^.5)^2
77.44 = y