Profit Maximisation Under Combined Constraints
A company produces a product and faces various production and market constraints. The revenue function is given by \(R(q)=150q-3q^2\) and the cost function by \(C(q)=20q+400\), where \(q\) represents the production level (in dozens of units). The profit function is \(P(q)=R(q)-C(q)\). Determine the production level \(q\) that maximises the profit, considering that \(q\) is restricted to […]
Part a
Using the functions provided, first write the profit function for the original cost structure. Given \(R(q)=150q-3q^2\) and \(C(q)=20q+400\), the profit function becomes \(P(q)=150q-3q^2-(20q+400)=130q-3q^2-400\). Differentiate \(P(q)\) with respect to \(q\), set the derivative equal to zero, and solve to find the optimal production level within the domain \([0,25]\).