Section 3.7 – Optimization Problems

Optimization Procedure 1.Draw a figure (if appropriate) and label all quantities relevant to the problem. 2.Focus on the quantity to be optimized. Name it. Find a formula for the quantity to be maximized or minimized. 3.Use conditions in the problem to eliminate variables in order to express the quantity to be maximized or minimized in terms of a single variable. 4.Find the practical domain for the variables in Step 3; that is, the interval of possible values determined from the physical restrictions in the problem. 5.If possible, use the methods of calculus to obtain the required optimum value (typically a max/min on an interval).

Example 1 A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river (He needs no fence along the river). What are the dimensions of the field that has the largest area? y Draw a Picture What needs to be Optimized? Area needs to be maximized: Eliminate Variable(s) with other Conditions Find the Domain: Use Calculus to Solve the Problem 600ft x 1200 ft xx River x-intercepts (no negative area)

Example 2 Find a positive number such that the sum of the number and its reciprocal is as small as possible. NOT APPLICABLE Draw a Picture What needs to be Optimized? The sum of a number and its reciprocal needs to be minimized: Eliminate Variable(s) with other Conditions Find the Domain: Use Calculus to Solve the Problem 1 THE EQUATION ALREADY HAS ONLY TWO VARIABLES 1 Since this is not a closed interval, check to make sure the critical point is a minimum. It is a minimum!

Example 3 A carpenter wants to make an open-topped box out of a rectangular sheet of tin 24 in. by 45 in. The carpenter plans to cut congruent squares out of each corner of the sheet and then bend and solder the edges of the sheet upward to firm the sides of the box. For what dimensions does the box have the greatest possible volume? x x 45 Draw a Picture What needs to be Optimized? Volume needs to be maximized: Eliminate Variable(s) with other Conditions Find the Domain: Use Calculus to Solve the Problem 5 in x 14 in x 35 in x x – 2x 45 – 2x x-intercepts (no negative volume)

Example 4 We need to design a cylindrical can with radius r and height h. The top and bottom must be made of copper, which will cost 2¢/in 2. The curved side is to be made of aluminum, which will cost 1 ¢/in 2. We seek the dimensions that will maximize the volume of the can. The only constraint is that the total cost of the can is to be 300 π cents. h Draw a Picture What needs to be Optimized? Volume needs to be maximized: Eliminate Variable(s) with other Conditions Find the Domain: Use Calculus to Solve the Problem r = 5 in & h= 20 in r x-intercepts (no negative volume)

Example 5 A liquid antibiotic manufactured by a pharmaceutical firm is sold in bulk at a price of $200 per unit. If the total production cost (in dollars) for x units is: C(x) = 500, x x 2 And if the production capacity of the firm is at most 30,000 units in a specified times, how many units of antibiotic must be manufactured and sold in that time to maximize the profit? NOT APPLICABLE Draw a Picture What needs to be Optimized? The profit needs to be maximized: Eliminate Variable(s) The equation already has one variable Find the Domain: Use Calculus to Solve the Problem 20,000 Units