4.4 Modeling and Optimization

1. 4.4 Modeling and Optimization

2. Using the Strategy • Find two numbers whose sum is 20 and whose product is as large as possible. • If one number is x, the other is (20 – x), and their product is f(x) = x(20 – x). • We can see from the graph of f in Figure 4.35 that there is a maximum. • From what we know about parabolas, the maximum occurs at x = 10. • The two numbers we see are x = 10 and 20 – x = 10.

3. Inscribing Rectangles • A rectangle is to be inscribed under one arch of the sine curve. What is the largest area the rectangle can have, and what dimensions give that area?

4. Inscribing Rectangles • Let (x , sin x) be the coordinates of point P in Figure 4.36. • From what we know about the sine function, the x-coordinate of point Q is . • Thus, = length of rectangle and sin x = height of rectangle. The area of the rectangle is See Example 2 on p. 220.

5. Examples from Business and Industry • To optimize something means to maximize or minimize some aspect of it. • What is the size of the most profitable production run? • What is the lease expensive shape for an oil can? • What is the stiffest rectangular beam we can cut from a 12-inch log? • We usually answer such questions by finding the greatest or smallest value of some function that we have used to model the situation.

6. Fabricating a Box • An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20- by 25-inch sheet of tin and bending up the sides (Figure 4.38). How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume?

7. Fabricating a Box • The height of the box is x, and the other two dimensions are (20 – 2x) and (25 – 2x). • Thus, the volume of the box is: V(x) = x(20 – 2x)(25 – 2x) • Because 2x cannot exceed 20, we have0 ≤ x ≤ 10. • Figure 4.39 suggests that the maximum value of V is about 820.53 and occurs at

8. Fabricating a Box • Expanding, we obtain V(x) = 40x³ - 90x² + 500x. The first derivative of V is: V’(x) = 12x² - 180x + 500. The two solutions of the quadratic equation V’(x)=0 are:

9. Fabricating a Box • Only c1 is in the domain [0 , 10] of V. The values of V at this one critical point and the two endpoints are: Critical point value: Endpoint values: Cutout squares that are about 3.68 in. on a side give the maximum volume, about 820.53 in.3

10. Designing a Can • See example 4 on p. 221 – 223.

11. Examples from Economics • Two places where calculus makes a contribution to the economic theory. • The first has to do with maximizing profit. • The second has to do with minimizing average cost.

12. Examples from Economics • Suppose that: r(x) = the revenue from selling x items c(x) = the cost of producing the x items p(x) = r(x) – c(x) = the profit from selling x items. • The marginal revenue, marginal cost, and marginal profit at this production level (x items) are:

13. Maximum Profit • Maximum profit (if any) occurs at a production level at which marginal revenue equals marginal cost.

14. Maximizing Profit • Suppose that r(x) = 9x and c(x) = x3 – 6x2 + 15x, where x represents thousands of units. Is there a production level that maximizes profit? If so, what is it?

15. Maximizing Profit • The possible production levels for maximum profit are approximately 0.586 thousand units or approximately 3.414 thousand units. • The graphs in Figure 4.43 show that maximum profit occurs at about x = 3.414 and maximum loss occurs at about x = 0.586. • Another way to look for optimal production levels is to look for levels that minimize the average cost of the units produced.

16. Minimizing Average Cost • The production level (if any) at which average cost is smallest is a level at which the average cost equals the marginal cost.

17. Minimizing Average Cost • Suppose c(x) = x3 – 6x2 + 15x, where x represents thousands of units. Is there a production level that minimizes average cost? If so, what is it?

18. Minimizing Average Cost • Since x > 0, the only production level that might minimize average cost is x = 3 thousand units. • We use the second derivative test. • The second derivative is positive for all x > 0, so x = 3 gives an absolute minimum.

19. More Practice!!!!! • Homework – Textbook p. 226 – 227