Honors Calculus 4.8. Optimization

Optimization To determine the optimum “Best!” value for various things.

Realize We optimize a lot all the time! Lowest cost, shortest distance, minimum total time, maximum area, etc. Realize: all these optimizations have maximums/minimums! That’s were the calculus can come in!

Steps to Optimization: 1. Write a Primary Equation What are you trying to optimize? 2. Write a secondary Equation What else do you know? 3. Use the secondary equation to re-write the primary equation so that it contains only one variable! 4. What is the feasible domain? Make sure values don’t get negative if it doesn’t make sense! 5. Find the absolute maximum/minimum Derivative, critical values AND endpoints

Some equations that may be helpful Boxes: SA: 2 𝑙𝑤 +2 𝑤ℎ +2 𝑙ℎ V: 𝑙𝑤ℎ Circles A: 𝜋 𝑟 2 Circumference: 2𝜋𝑟

Example One (The Box) A manufacturer wants to design an open box with a square base and a surface area of 108 square inches. What dimensions will produce a box with the maximum volume? Primary Equation Secondary Equation Re-Write Primary Domain? Absolute Max/Min!

Example Two: An open box is to be made from a square piece of material, 24 inches wide, by cutting equal sized squares from the corners and turning up the sides (see the picture below). Determine the area of the squares on the side in order to maximize the volume of the box. Primary Equation Secondary Equation Re-Write Primary Domain? Absolute Max/Min!

Example Three: A box with a square base and open top must have a volume of 42,592 𝑐 𝑚 3 . Find the dimensions of the box that minimize the amount of material used to create the box. Primary Equation Secondary Equation Re-Write Primary Domain? Absolute Max/Min!

You Try! According to U.S. Postal regulations the girth plus the length of the parcel sent by mail may not exceed 108 inches, where by “girth” we mean the perimeter of the smallest end. What is the largest possible volume of a rectangular parcel with a square end that can be sent by mail? What are the dimensions of the package of largest volume?

Example Four: (2D shapes) Maximize the area of a rectangle if the rectangle has a perimeter of 100 meters.

Example Five: Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much of the wire should be used for the circle to enclose the maximum total area?

Do Now: We need to enclose a rectangular field with a fence. We want to enclose 10,000 square feet. A building is on one of the longer sides of the field and so won’t need any fencing. Determine the dimensions of the field that will require the least amount of fencing.

Example Six: A rancher has 200 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?

Example Seven: A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1.5 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

Example Eight: A Norman window has the outline of a semicircle on top of a rectangle as shown in the figure. Suppose there is 8+𝜋 feet of wood trim available for all four sides of the rectangle and semicircle. Find the dimensions of the rectangle (and hence, the semicircle) that will maximize the area of the window.

Answers to the You Try! 1. 𝑥=1.94 𝑓𝑡 , 𝑦=5.15 ft (if you labeled x and y differently than me– you may have the opposite. That’s fine!) 2. R and H are both 1.68 meters

You Try! The overall area you are working with is 60 square feet and you want to divide the area up into six pens of equal size as shown below. The cost of the outside fencing is $10 a foot and the inside fencing costs $5 a foot. Your goal is to minimize the cost of the fencing. What are the dimensions of each pen that will minimize the cost of the breeding ground?

You Try #2: A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the most light? (Realize you are maximizing the area). Round to the nearest hundredth. Note: Area of a Circle = 𝜋 𝑟 2 and Circumference of a circle =2𝜋𝑟

Example Eight (Slightly Different 2D) Find the largest rectangular area that can be inscribed under the curve 𝑓 𝑥 =16− 𝑥 2 when bounded by the x axis.

Example Nine: A rectangle is bounded by the x and y axes and the graph of 𝑦= 6−𝑥 2 . What length and width should the rectangle have such that its area is a maximum?

You Try! A rectangle is bounded by the x axis and the semicircle 𝑦= 25− 𝑥 2 . What length and width should the rectangle have so that its area is a maximum?

Example Ten (Find 2 numbers…) Find two positive numbers such that their sum is 60 and the product of one number and the square of the other number is maximized.

Example Eleven: Find two positive numbers such that the second number is the reciprocal of the first and the sum is a minimum.

You Try! Find two positive numbers such that the sum of the first and twice the second is 100 and the product is a maximum.

Example Twelve (Pythag!) Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed (supported) by two wires, attached at 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?

Example Thirteen: The figure below is a rectangle and an isosceles right triangle. The perimeter is 20 inches. What dimensions (length and width of the rectangle) maximum the total area?

You Try! A triangle is inscribed in a semicircle of radius 10 cm so that one side is along the diameter. Find the dimension of the triangle with maximum area.

Example Fourteen: (Distance!) Recall: The distance formula is d= x 2 − x 1 2 + y 2 − y 1 2 For these questions, go from the given point to any point (x, y). Which points on the graph of 𝑦=4− 𝑥 2 are closest to the point (0, 2)?

Example Fifteen: Which points on the graph of 𝑦= 𝑥−8 are closest to the point 2, 0 ?

You Try! Find the point on 𝑓 𝑥 = 4𝑥 closest to (4, 0) Find the point on the graph of 𝑦= 𝑥+8 closest to the point (2, 0)

Practice Problems Try some on your own/in your table groups As always don’t hesitate to ask me questions if you are confused!