Luigi owns a burrito truck called \"El Guapo\". He uses two inputs (labour and capital) to make his only product, carnitas burritos, which he sells on campus.
Denote $x_1$ as the amount of labour he uses, and $x_2$ as the amount of capital. His production function for burritos is: $f(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{3}}$.
He sells his burritos for $10, and in the short-run, his capital is fixed at 8. If the price of labour is w, and the price of capital is r, what is the firm's demand for labour in the short run?