What combination of inputs results in the largest output? I'm trying to solve the following problem:

Production of a certain company depends on $3$ inputs $x,y,z$ in the following way:
$$f(x,y,z) = 50x^{\frac{2}{5}} y^{\frac{1}{5}} z^{\frac{1}{5}}$$
Budget of the company is $24 000$ dollars and products $x, y, z$ can be bought for $80$, $12$ or $10$ dollars per unit in that order. What combination of inputs results in biggest production?

It's a Calculus test preparation problem and I have no idea how I would start. Could you help me?
My idea is to use constrained extremas but I don't know how.

By AM-GM $$24000=80x+12y+10z=2(20x+20x+6y+5z)\geq8\sqrt[4]{20^2\cdot6\cdot5x^2yz}.$$
The equality occurs for $20x=6y=5z$ or $(x,y,z)=(150,500,600)$.
I.e., $x^2yz$ (and so $50(x^2yz)^{\frac{1}{5}}$) attains a maximal value for $$(x,y,z)=(150,500,600).$$ 
