Production function is: $f(L,M)=L^{1/2}M^{1/2}$. L is the number of units of labour, M of machines used. Cost of labour is 9 per unit, whereas the cost of machine is 81 per unit. Total cost of producing 10 units of output will be:
a) 270
b) 90
c) 135
d) 450
e) non of the above
Why its is 135? I completely do not get it. What is the proper method?
The production function states the quantity that a firm can produce. So if it produces $10$ units:
$$10=f(L,M)=L^{1/2}M^{1/2}$$
Hence:
$$100=LM$$
$$\frac{100}{L}=M$$
We know the cost will be:
$$C=9L+81M$$
$$=9L+\frac{8100}{L}$$
Minimizing this using standard procedures $C'(L)=0$.. gives $L=30$ $C=540$ so I think you typed something wrong.
I have reason to believe that the actual production function is:
$$f(L,M)=4L^{1/2}M^{1/2}$$