Question

manufacturer of headphones is preparing to set the price on a new design. Demand is thought to depend on the price and is represented by the model D = 2,500 - 3P The accounting department estimates that the total costs can be represented by C = 5,000 + 5D Develop a model for the total profit in terms of the price, P.

Answer 1

To find the total profit model in terms of price P, you must calculate total revenue as the product of price and demand, then calculate total costs using the cost model. Subtract total costs from total revenue to obtain the profit model, and then simplify to express in terms of P.

Step-by-step explanation:

To develop a model for the total profit in terms of the price P, we need to first determine the total revenue and total costs. Total revenue (TR) is calculated by multiplying the price per unit (P) by the demand (D), so TR = P × D. Given the demand model D = 2,500 - 3P, we can write this as TR = P × (2,500 - 3P).

Next, to find total costs (TC), we use the cost model C = 5,000 + 5D. Substituting the demand model into the cost model, we get TC = 5,000 + 5 × (2,500 - 3P).

The profit model is the difference between total revenue and total costs, which is Profit = TR - TC. Substituting the TR and TC from above, this gives us Profit = P × (2,500 - 3P) - (5,000 + 5 × (2,500 - 3P)). By simplifying, we obtain the profit model in terms of P.