ISI 2013 eco doubt

[The Villain]

Please help me this question.
A firm is deciding whether to hire a worker for a day at a daily wage
of Rs. 20/-. If hired, the worker can work for a maximum of 10 hours
during the day. The worker can be used to produce two intermediate
inputs, 1 and 2, which can then be combined to produce a final good. If
the worker produces only 1, then he can produce 10 units of input 1 in
an hour. However, if the worker produces only 2, then he can produce
20 units of input 2 in an hour. Denoting the levels of production of
the amount produced of the intermediate goods by k1 and k2 , the
production function of the final good is given by √k1k2. Let the final
product be sold at the end of the day at a per unit price of Rs. 1/-.
Solve for the firms optimal hiring, production and sale decision.

[kangkan]

Hi..once the labour is hired..i have him for the the entire 10 hours..since the price is constant , and the once hired,the labour is a sunk cost, my problem is to maximize out put given the factor supply...

let me produce x1 of og good k1 and x2 of good k2... i will use use x1/10 and x2/20 hours of the labour respectively..hence my budget constraint is x/10 +x2/20 =10.

now my probel is max root over x1*x2 subject to x1/10+x2/20=10..apply lagrangean multipier..we get x1=50 and x2=100...at this inputs y=38..profit will 38.7-20>0..hence it will be profitable for me to hire the labour