Math 142 Section 5.6 Examples Example 1: Find two nonnegative numbers whose sum is 55 and whose product is minimal. Step 1: Identify the unknowns. Step 2: Identify the variable that is to be optimized and express it as a function of the others. Step 3: Find any relations between the variables and use that to obtain a function of one variable. Step 4: Determine the interval over which we are optimizing. Step 5: Optimize the function using our previous techniques. Math 142 Section 5.6 Examples Example 2: A farmer has 500 yards of fencing with which to fence in three sides of a rectangular pasture (a river forms the fourth side). Find the dimensions of the pasture with the greatest area that the farmer can fence. Example 3: A theater owner charges $5 per ticket and sells 250 tickets. Through market analysis, the owner learns that for every one dollar she raises the price, she will lose 10 customers. What should she charge to maximize revenue? Math 142 Section 5.6 Examples Example 4: A box with a square base and an open top must have a volume of 62500 cm3 . Find the dimensions of the box that minimize the amount of material used. Math 142 Section 5.6 Examples Example 5: A manufacturer is going to make rectangular containers with a square base and a volume of 128 cubic feet. The cost of materials for the top and four sides is $2 per square foot. The cost of material for the bottom, is $6 per square foot. Find the dimensions of the box that minimizes the cost of the materials.