Suppose a company has a production function of the form Q = KL, where Q is the quantity of output, K is the amount of capital, and L is the amount of labor. The price of capital is $10 per unit, and the price of labor is $5 per unit. The company's budget is $500, and it wants to produce as much output as possible.

a) What is the company's production function?
b) What is the equation for the company's budget constraint?
c) What is the marginal product of labor?
d) What is the marginal rate of technical substitution?
e) What is the optimal combination of capital and labor that maximizes output?

This question is worth 10 marks. Please show all your calculations and provide your answers in full sentences.