Section 4.14 : Business Applications

1. A company can produce a maximum of 1500 widgets in a year. If they sell x widgets during the year then their profit, in dollars, is given by, \[P\left( x \right) = 30,000,000 - 360,000x + 750{x^2} - \frac{1}{3}{x^3}\] How many widgets should they try to sell in order to maximize their profit? Solution
2. A management company is going to build a new apartment complex. They know that if the complex contains x apartments the maintenance costs for the building, landscaping etc. will be, \[C\left( x \right) = 4000 + 14x - 0.04{x^2}\] The land they have purchased can hold a complex of at most 500 apartments. How many apartments should the complex have in order to minimize the maintenance costs? Solution
3. The production costs, in dollars, per day of producing x widgets is given by, \[C\left( x \right) = 1750 + 6x - 0.04{x^2} + 0.0003{x^3}\] What is the marginal cost when \(x = 175\) and \(x = 300\)? What do your answers tell you about the production costs? Solution
4. The production costs, in dollars, per month of producing x widgets is given by, \[C\left( x \right) = 200 + 0.5x + \frac{{10000}}{x}\] What is the marginal cost when \(x = 200\) and \(x = 500\)? What do your answers tell you about the production costs? Solution
5. The production costs, in dollars, per week of producing x widgets is given by, \[C\left( x \right) = 4000 - 32x + 0.08{x^2} + 0.00006{x^3}\] and the demand function for the widgets is given by, \[p\left( x \right) = 250 + 0.02x - 0.001{x^2}\] What is the marginal cost, marginal revenue and marginal profit when \(x = 200\) and \(x = 400\)? What do these numbers tell you about the cost, revenue and profit? Solution