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
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
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
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
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