7. Optimization

Optimization Any time you use a superlative case – biggest, smallest, cheapest, strongest, best, least, ugliest, etc – you are trying to optimize What keeps us from making something infinitely big, infinitely small,…..? There is some limiting factor, or constraint, that prevents that from happening

Strategy Draw a picture – label unknowns with variables. Label constants as numbers. Write a primary equation for the quantity to be optimized Identify the limiting factor and write a secondary equation (usually) involving this constraint. Solve the second equation for any convenient variable and plug into the primary equation to establish an equation for the optimal quantity in terms of a single variable Find critical values Determine the absolute extrema (remember to check endpoints if given an interval!) May need to eliminate an answer that doesn’t make sense

Example 1 Find 2 numbers whose sum is 60 and the product of one times the square of the other is as large as possible.

Example 2 A rectangular page is to contain 24 square inches of print. The top and bottom margins are to be 1½ in, the left and right margins are to be 1 in. What should dimensions be so least amount of paper is used?

Example 3 An open top box is to be made by cutting congruent squares of side length x from the corners of a 20x25 in sheet of tin and bending up the sides. How large should the squares be to make the box to hold as much as possible? What is resulting maximum volume?

Example 4 You have been asked to design a 1000 cm3 oil can shaped like a cylinder. What dimensions will use the least material?

Example 5 We need to enclose a portion of a field with a rectangular fence. We have 500 feet of fencing material and 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 enclose the largest area.

Example 6 - Minimizing travel time Clint wants to build a dirt road from his ranch to the highway so he can drive to the city in the least amount of time. The perpendicular distance to the highway is 4 miles and the city is 9 miles down the highway. Speed limit is 20 mph on dirt road and 55 mph on highway.

Example 7 Determine the point(s) on y = x2 + 1 that are closest to (0, 2).

Example 8 A 2 foot piece of wire is cut into 2 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 is a maximum?

Example 9 The manager of an 80-unit apartment complex is trying to decide what rent to charge. Experience has shown that at a rent of $200, all of the units will be full. On the average, one additional unit will remain vacant for each $20 increase in rent. Find the rent to charge to maximize revenue.

Example 10 Find an equation for the tangent line to the graph of having maximum slope