Section 3 7 Optimization Problems Optimization Procedure 1

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 River x Draw a Picture x y What needs to be Optimized? Area needs to be maximized: x-intercepts (no negative area) 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? Find the Domain: Use Calculus to Solve the Problem Eliminate Variable(s) with other Conditions 600 ft x 1200 ft

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

Example 3 24 – 2 x Draw a Picture 45 – 2 x x x 24 x-intercepts (no negative volume) 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? Find the Domain: Use Calculus to Solve the Problem 45 What needs to be Optimized? Volume needs to be maximized: Eliminate Variable(s) with other Conditions 5 in x 14 in x 35 in

Example 5 Draw a Picture r h What needs to be Optimized? Volume needs to be maximized: x-intercepts (no negative volume) 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 300π cents. Find the Domain: Use Calculus to Solve the Problem Eliminate Variable(s) with other Conditions r = 5 in & h= 20 in

Example 4 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, 000 + 80 x + 0. 003 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? Draw a Picture NOT APPLICABLE Find the Domain: Use Calculus to Solve the Problem What needs to be Optimized? The profit needs to be maximized: Eliminate Variable(s) The equation already has one variable 20, 000 Units