Differential Equations in Economics

The equilibrium price is the price at which the quantity demanded equals the quantity supplied. In this case, the quantity supplied is constant, so the equilibrium price is the price that makes the quantity demanded equal to the constant quantity supplied.

The marginal revenue is the derivative of the total revenue function. So, (MR = \frac{dR}{dQ} = p - Q). When (Q = 10), (MR = p - 10). Since the price is not given in the question, we cannot find the exact value of the marginal revenue.

To find the population size after 10 years, we need to solve the differential equation. The solution is (P(t) = P_0e^{kt}), where (P_0) is the initial population size, (k) is the growth rate, and (t) is the time. In this case, (P_0 = 100) and (k = 0.5). So, after 10 years, the population size will be (P(10) = 100e^{0.5 * 10} = 300).

To find the equilibrium point, we need to solve the system of differential equations. The equilibrium point is the point where the population sizes of both predators and prey are constant. In this case, the equilibrium point is ((P, Q) = (200, 400)).

The average cost is given by the function (AC(Q) = \frac{C(Q)}{Q}). When (Q = 10), (AC(10) = \frac{C(10)}{10} = \frac{0.5(10)^2 + 2(10) + 10}{10} = 4.50).

The equilibrium quantity is the quantity at which the quantity supplied equals the quantity demanded. In this case, the quantity demanded is constant, so the equilibrium quantity is the quantity that makes the quantity supplied equal to the constant quantity demanded.

The marginal cost is the derivative of the total cost function. So, (MC = \frac{dC}{dQ} = 0.75Q^2 + 3Q + 4). When (Q = 10), (MC = 0.75(10)^2 + 3(10) + 4 = 11.75).

To find the population size after 10 years, we need to solve the differential equation. The solution is (P(t) = P_0e^{kt}), where (P_0) is the initial population size, (k) is the growth rate, and (t) is the time. In this case, (P_0 = 100) and (k = -0.2). So, after 10 years, the population size will be (P(10) = 100e^{-0.2 * 10} = 6.25).

To find the equilibrium point, we need to solve the system of differential equations. The equilibrium point is the point where the population sizes of both predators and prey are constant. In this case, the equilibrium point is ((P, Q) = (200, 400)).

The marginal revenue is the derivative of the revenue function. So, (MR = \frac{dR}{dQ} = 10 - Q). When (Q = 10), (MR = 10 - 10 = 0).

The equilibrium price is the price at which the quantity demanded equals the quantity supplied. In this case, the quantity supplied is constant, so the equilibrium price is the price that makes the quantity demanded equal to the constant quantity supplied.

The average cost is given by the function (AC(Q) = \frac{C(Q)}{Q}). When (Q = 10), (AC(10) = \frac{C(10)}{10} = \frac{0.1(10)^3 + 0.5(10)^2 + 2(10) + 10}{10} = 4.60).

To find the population size after 10 years, we need to solve the differential equation. The solution is (P(t) = P_0e^{kt}), where (P_0) is the initial population size, (k) is the growth rate, and (t) is the time. In this case, (P_0 = 100) and (k = 0.4). So, after 10 years, the population size will be (P(10) = 100e^{0.4 * 10} = 260).

To find the equilibrium point, we need to solve the system of differential equations. The equilibrium point is the point where the population sizes of both predators and prey are constant. In this case, the equilibrium point is ((P, Q) = (200, 400)).