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### Section 4-14 : Business Applications

1. A company can produce a maximum of 2500 widgets in a year. If they sell x widgets during the year then their profit, in dollars, is given by, $P\left( x \right) = 500,000,000 - 1,540,000x + 1450{x^2} - \frac{1}{3}{x^3}$ How many widgets should they try to sell in order to maximize their profit?
2. A company can produce a maximum of 25 widgets in a day. If they sell x widgets during the day then their profit, in dollars, is given by, $P\left( x \right) = 3000 - 40x + 11{x^2} - \frac{1}{3}{x^3}$ How many widgets should they try to sell in order to maximize their profit?
3. 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, $C\left( x \right) = 70,000 + \frac{{2736}}{5}x - \frac{{211}}{{50}}{x^2} + \frac{1}{{150}}{x^3}$ The land they have purchased can hold a complex of at most 400 apartments. How many apartments should the complex have in order to minimize the maintenance costs?
4. The production costs of producing x widgets is given by, $C\left( x \right) = 2000 + 4x + \frac{{90,000}}{x}$ If the company can produce at most 200 widgets how many should they produce to minimize the production costs?
5. The production costs, in dollars, per day of producing x widgets is given by, $C\left( x \right) = 400 - 3x + 2{x^2} + 0.002{x^3}$ What is the marginal cost when $$x = 20$$ and $$x = 75$$? What do your answers tell you about the production costs?
6. The production costs, in dollars, per month of producing x widgets is given by, $C\left( x \right) = 10,000 + 14x - \frac{{8,000,000}}{{{x^2}}}$ What is the marginal cost when $$x = 80$$ and $$x = 150$$? What do your answers tell you about the production costs?
7. The production costs, in dollars, per week of producing x widgets is given by, $C\left( x \right) = 65,000 + 4x + 0.2{x^2} - 0.00002{x^3}$ and the demand function for the widgets is given by, $p\left( x \right) = 5000 - 0.5x$ What is the marginal cost, marginal revenue and marginal profit when $$x = 2000$$ and $$x = 4800$$? What do these numbers tell you about the cost, revenue and profit?
8. The production costs, in dollars, per week of producing x widgets is given by, $C\left( x \right) = 800 + 0.008{x^2} + \frac{{56,000}}{x}$ and the demand function for the widgets is given by, $p\left( x \right) = 350 - 0.05x - 0.001{x^2}$ What is the marginal cost, marginal revenue and marginal profit when $$x = 175$$ and $$x = 325$$? What do these numbers tell you about the cost, revenue and profit?