Calculus 220 section 2 5 Optimization Problems Applications notes by Tim Pilachowski Oh boy now we get to start on word problems First a definition optimize verb to make as perfect or effective as possible In calculus terms for anything optimal we will be searching for some sort of maximum or minimum Example A The function s t 16t 2 10t 240 calculates the height of an object s after time t thrown upward at 10 feet per second from a bridge which is 240 feet above the river below In section 1 8 lecture notes we asked a What is the height of the rock after 2 seconds answer Find s 2 196 feet b How much height did the rock gain after 2 seconds answer Calculate s 2 s 0 loss of 44 feet of height c What is the velocity of the rock after 2 seconds answer Find s 2 the rock is falling at 54 feet per second s 2 s 0 d What is the average velocity during the first 2 seconds answer Find on average the rock has 2 0 fallen at 22 feet per second New questions e How long does it take for the rock to reach its maximum height Answer 5 sec 16 f What is the maximum height above the water reached by the rock Answer 241 9 feet 16 g What is the maximum height above the bridge reached by the rock Answer 1 9 feet 16 Example B Public health officials use rates of change to quantify the spread of an epidemic into an equation which they then use to determine the most effective measures to counter it A recent measles epidemic followed the equation y 45t 2 t 3 where y the number of people infected and t time in days a What is the domain of this function Answer 0 t 45 days b How many people are infected after 5 days Answer 1000 people c What is the rate of spread after 5 days Answer 375 new cases per day d After how many days does the number of cases reach its maximum Answer 30 days e Use the above to sketch the graph of y Example C Perhaps you have already encountered versions of the infamous corral problem favored by Math 113 and 220 course coordinators A farmer has 900 feet of fencing with which to y overall length build a pen for his animals and being a frugal sort doesn t want to buy any more fencing He needs two pens but can build them adjacent to each other sharing one x side as in the diagram to the left Find the dimensions that will give the maximum area Answer 225 ft long by 150 ft wide width Example D Optimization does not always involve a maximum The fuel maintenance and labor costs in dollars per mile of operating a truck on an interstate highway are described as a function of the truck s velocity miles per hour by the algebraic rule C v 78 1 2v 5880v 1 What speed should the driver maintain on a 600 mile haul to minimize costs Answer 70 mph
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