3 Many times in life we are asked to do an optimization problem – that is, find the largest or smallest value of some quantity…Find the route which will minimize the time it takes me to get to schoolBuild a structure with the least amount of materialBuild a structure costing the least amount of moneyFind the least medication one should take to help with a medical problem.Find the most a company should charge for a CD in order to make as much money as possible.Build a yard enclosure with the most amount of space.
4 A company that manufactures bicycles estimates that the profit for selling a particular model is P(t)= -45x x ,000 where P is the profit (in dollars) and x is the advertising expense (in tens of thousands of dollars). According to this model, how much money should they spend on advertising to maximize their profit?
5 Find the volume of a box that can be made by cutting out squares from the corners of an 8 inch by 15 inch rectangular sheet of cardboard and folding up the sides. Justify your answer.
6 If a farmer has 250 feet of fence and wants to make a rectangular pen with a divider to separate animals, one side of which is along the side of his barn, what dimensions should be used in order to maximize the area of the pen?
7 Summary of the steps…Step 1: Name the variables (there are usually 2) and write down any relationship between the variables (this is the secondary equation).Step 2: Write down the function that is to be maximized or minimized, giving it a name (this is the primary equation).Step 3: Reduce the number of variables in the function you are maximizing or minimizing (primary equation) to one variable using the relation written in step 1. Completely simplify this equation!Step 4: Find the critical number(s) of your function and verify whether they give max or min using the first derivative number line and answer the question.
8 A soccer field will be fenced in at McAlpine Park in the near future A soccer field will be fenced in at McAlpine Park in the near future. No fence will be required on the side lying along the creek. If the new wood fence costs $12 per meter for the side parallel to the creek, and $4 per meter for the other two sides, find the dimensions of the soccer field of maximum area that could be enclosed with a budget of $3,600.
9 If a closed can with volume 24π cubic inches is made in the form of a cylinder, find the height and radius of the can with minimum surface area. (V= πr2h, A= 2πr2 + 2πrh)
10 A rectangular box is to be made with a square base and having a volume of 100 in3. If the cost of the materials are $2 per square inch for the bottom, $1.50 per square inch for the sides and $4 per square inch for the top, what dimensions will yield the least expensive box?