Could someone please check my work on the following problem?
"The quantity, q, of a product manufactured depends on the number of workers, W, and the amount of capital invested, K, and is given by
q = 6 * W^(3/4) * K^(1/4)
Labor costs are $10 per worker and capital costs are $20 per unit, and the budget is $2000.
What are the optimum number of workers and the optimum number of units of capital?"
And here's my work:
Budget/constraint = 10W+20K = 2000, so
Minimize q subject to 10W+20K = 2000
q= 6 * W^(3/4) * K^(1/4) + λ(10W+20K - 2000)
∂q/∂W = 6 (3/4) W^(-1/4) k^(1/4) + 10λ = 0 -------(1)
∂q/∂K = 6 W^(3/4) (1/4) K^(-3/4) + 20λ = 0 -------(2)
18K^(1/4) / 4W^(1/4) +10 λ = 0 -------(1)
6 W^(3/4) /4K^(3/4) + 20 λ = 0 -------(2)
Eq. 1 multiplied by eq. 2
36 K^(1/4)/4W^(1/4) + 20 λ = 0 -------(1)
6 W^(3/4) /4K^(3/4) + 20 λ = 0 -------(2)
20 λ = -36 K^(1/4)/4W^(1/4)
20 λ = -6 W^(3/4) /4K^(3/4)
-36 K^(1/4)/4W^(1/4) = -6 W^(3/4) /4K^(3/4)
6 K^(1/4)/W^(1/4) = W^(3/4) /K^(3/4)
W=6K
10W+20K = 2000
60K+20K=2000
80K=2000
K=200/8 = 25
10W+500 = 2000
10W=1500
W=150
W=150
K=25