Given the production function
$Q := \sqrt K + L^2$,
determine the optimal level of production and the relative demand of the two inputs capital $K$ and work $L$.
The cost of a unit of capital is $r = 5$, the cost of a unit of work is $w = 60$, the unit price of the product is $p = 10$.
My book gives the following reference solution: optimal solution: $$K = 1, L = 3, Q = 10.$$
For what concerns the solution, I can say that the cost function is the following:
$$C = 5K + 60L.$$
For the rest, I do not know how to use the pricing information $p = 10$.
Could anyone help me?
Thank you for considering my request.
A profit maximising firm seeks to maximize the following objective $$\max_{K,L}\{\Pi=p\cdot Q-TC\}$$ where $\Pi$,$pQ$ and $TC$ are the profits, revenue and the total cost respectively. Note that $TC=rK+wL$. Subsitute $Q=\sqrt{K}+L^2$ and rewrite the obejctive function as $$\max_{K,L}\{\Pi=p\cdot (\sqrt{K}+L^2)-(rK+wL)\}$$ First order conditions wrt $K$ and $L$ are respectively $$\frac{\partial{\Pi}}{\partial{K}}=\frac{p}{2\sqrt{K}}-r=0$$ and $$\frac{\partial{\Pi}}{\partial{L}}=2pL-w=0$$ These two equations will yield the optimal values for capital and labour as $$K=(\frac{p}{2r})^2=(\frac{10}{2\cdot 5})^2=1$$ and $$L=\frac{w}{2p}=\frac{60}{2\cdot 10}=3$$ And you are done.