AB Lesson 5.4 Optimization To optimize: 1.) Choose variables (draw a diagram to assign appropriate variables) 2.) Find the function and the interval (write a function in one variable using information in problem) 3.) Optimize the function How to optimize (find the max or min): 1.) Find the critical points of the function on [a, b] 2.) Evaluate the function at the critical points and the endpoints a and b 3.) The largest and smallest of these values are the extreme values Ex. 1 – A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 245,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river? You Try – A rancher has 400 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum? Ex. 2 – Which points on the graph of 𝑓 𝑥 = 4 − 𝑥 ! are closest to the point (0, 2)? You Try – Which point on the graph of 𝑓 𝑥 = 𝑥 + 2 are closest to the origin? Ex. 3 – The sum of two numbers is 14, what values of x and y will maximize the product of the two numbers? You Try – The product of two numbers is 36, what values of x and y will maximize the expression 2𝑥 + 𝑦? Ex. 4 – Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 337.5 square centimeters. Ex. 5 – Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire? Ex. 6 – A Norman window is constructed by adjoining a semicircle to the top of an ordinary window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet. Plickers (Calc) – A 2 feet piece of wire is cut into two pieces and one piece is bent into a square and the other is bent into an equilateral triangle. Where should the wire be cut so that the total area enclosed by both shapes is at the minimum? (A) 0.870 (B) 0.846 (C) 0.827 (D) 0.8112 Homework: Pg.230 #1-­‐11 Pg.231 #16,19-­‐21,23,53-­‐56