Suppose a country has a production possibility frontier given by the equation Q = 10L^0.5K^0.5, where Q is the quantity of output, L is the quantity of labor, and K is the quantity of capital. The country's endowment of labor is 400 units, and its endowment of capital is 900 units.

a) Derive the equation for the country's production possibility frontier, and graph the frontier using L as the horizontal axis and Q as the vertical axis.

b) Suppose the country is in autarky, and the wage rate is 10 and the rental rate of capital is 20. Derive the equation for the country's autarky relative price of labor in terms of the relative price of capital, and graph the equation using the relative price of labor as the horizontal axis and the relative price of capital as the vertical axis.

c) Now suppose the country opens up to trade and can trade with another country that has a relative price of labor of 1.5. If the country chooses to specialize in the production of the good that uses relatively more labor, what will be its relative price of labor in terms of the relative price of capital in the world market? Explain your answer.

d) Derive the country's production and consumption after trade, given that the world relative price of labor is 1.5. Show that the country has gained from trade and calculate its gains from trade.