In a simple economic model, the demand for a product is given by the differential equation (\frac{dQ}{dt} = -aQ + bP), where (Q) is the quantity demanded, (P) is the price, (a) and (b) are positive constants. What is the equilibrium price?

A company's total revenue is given by the function (R(Q) = pQ - \frac{1}{2}Q^2), where (Q) is the quantity produced and (p) is the price per unit. The marginal revenue is given by the derivative of the total revenue function. What is the marginal revenue when (Q = 10) units?

A population grows according to the differential equation (\frac{dP}{dt} = 0.5P), where (P) is the population size. If the initial population size is 100, what will be the population size after 10 years?

In a predator-prey model, the population of predators is given by the differential equation (\frac{dP}{dt} = 0.5P - 0.2PQ), where (P) is the population of predators and (Q) is the population of prey. The population of prey is given by the differential equation (\frac{dQ}{dt} = -0.3Q + 0.1PQ). What is the equilibrium point of this system?

A company's cost function is given by the function (C(Q) = 0.5Q^2 + 2Q + 10), where (Q) is the quantity produced. The average cost is given by the function (AC(Q) = \frac{C(Q)}{Q}). What is the average cost when (Q = 10) units?

In a simple economic model, the supply of a product is given by the differential equation (\frac{dQ}{dt} = aP - bQ), where (Q) is the quantity supplied, (P) is the price, (a) and (b) are positive constants. What is the equilibrium quantity?

A company's total cost function is given by the function (C(Q) = 0.25Q^3 + 1.5Q^2 + 4Q + 10), where (Q) is the quantity produced. The marginal cost is given by the derivative of the total cost function. What is the marginal cost when (Q = 10) units?

A population grows according to the differential equation (\frac{dP}{dt} = -0.2P), where (P) is the population size. If the initial population size is 100, what will be the population size after 10 years?

In a predator-prey model, the population of predators is given by the differential equation (\frac{dP}{dt} = -0.3P + 0.1PQ), where (P) is the population of predators and (Q) is the population of prey. The population of prey is given by the differential equation (\frac{dQ}{dt} = 0.2Q - 0.05PQ). What is the equilibrium point of this system?

A company's revenue function is given by the function (R(Q) = 10Q - 0.5Q^2), where (Q) is the quantity sold. The marginal revenue is given by the derivative of the revenue function. What is the marginal revenue when (Q = 10) units?

In a simple economic model, the demand for a product is given by the differential equation (\frac{dQ}{dt} = -2Q + 10P), where (Q) is the quantity demanded, (P) is the price, and (a) and (b) are positive constants. What is the equilibrium price?

A company's total cost function is given by the function (C(Q) = 0.1Q^3 + 0.5Q^2 + 2Q + 10), where (Q) is the quantity produced. The average cost is given by the function (AC(Q) = \frac{C(Q)}{Q}). What is the average cost when (Q = 10) units?

A population grows according to the differential equation (\frac{dP}{dt} = 0.4P), where (P) is the population size. If the initial population size is 100, what will be the population size after 10 years?

In a predator-prey model, the population of predators is given by the differential equation (\frac{dP}{dt} = 0.2P - 0.1PQ), where (P) is the population of predators and (Q) is the population of prey. The population of prey is given by the differential equation (\frac{dQ}{dt} = -0.3Q + 0.2PQ). What is the equilibrium point of this system?