Stuck on this Economics question:
Use calculus to determine the optimal levels of $L$ and $K$ that are required to maximize profit for this firm.
Useful information: A firm sells its output in a perfectly competitive market at a fixed price of $20$ per unit. It buys two inputs $L$ and $K$ at prices of $15$ and $50$ per unit respectively and has the following production function: $Q = 100\cdot L^{0.3}K^{0.5}$.
$\text{Profit} = 2000\cdot L^{0.3} K^{0.5} - 15L - 50K$.
This function$$f(L,K)=2000L^{0.3}K^{0.5}-15L-50K$$ has a maximum at the point $(L,K)$ where the gradient of the function is zero, i.e $$\frac{\partial f(L,K)}{\partial L}=\frac{600\sqrt{k}}{L^{0.7}}-15=0$$ and $$\frac{\partial f(L,K)}{\partial K}=\frac{1000L^{0.3}}{\sqrt{k}}-50=0.$$ When you solve this system of equations, you get the answer $$L=18 101933.59...$$ $$K=9050966.79...$$