Optimization (Max/Min)

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Presentation transcript:

Optimization (Max/Min) Honors Calculus Keeper 25

Strategies for Solving Max-Min Problems Read the problem carefully. If Relevant, draw a picture. Label the picture with appropriate variables and constants, noting what varies and what stays fixed. Translate the problems to an equation involving a quantity to be maximized or minimized. Represent the quantity in terms of the variables of Step 2. Try to express your quantity as a function of one variable. Use the derivative to determine the maximum or minimum values and the points at which they occur.

Example 1 From a thin piece of cardboard 8” x8”, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?

Example 2 Your friend asks you to produce a box from cardboard that is 30” x 30” with dimensions that will maximize volume. What is the maximum volume?

Example 3 A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposites will use heavy fencing selling for $3 a foot. While the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced at a cost of $6000.

Example A soup company is constructing an open-top, square based, rectangular metal tank that will have a volume of 32 cubic feet. What dimensions yield the minimum surface area? What is the minimum surface area?

example A stereo manufacturer determines that in order to sell x units of a new stereo, the price per unit must be 𝑝=1000− 𝑥. The manufacturer also determines that the total cost of producing 𝑥 units is given by 𝐶 𝑥 =3000+20𝑥. Find the total revenue 𝑅(𝑥). Find the total profit 𝑃 𝑥 . How many units must the company produce and sell in order to maximize profit? What is the maximum profit? What price per unit must be changed in order to make the maximum profit?

Example A university is trying to determine what price to charge for football tickets. At a price of $6 per ticket, it averages 70,000 people per game. For every increase of $1, it loses 10,000 people from the average number. Every person at the game spends an average of $1.50 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?