MATH-1311 - Brucks - Fall 2016 Name: ____________________________________________ Optimization Project Instructions: Answer all of the following on your own paper. Show all necessary work neatly, using proper notation, and box your answers. Unless otherwise noted, exact answers are expected, as opposed to decimal approximations. Students may collaborate with classmates on this project, but are expected to seek or receive NO outside assistance, and are responsible for their own, individual submission. We have learned that derivatives can be used to find extreme values (max/min) of functions, and in this project we apply that knowledge in the context of geometric, physical, and financial problems. These are called optimization problems, because our goal is to find the \"best\" value of a function, whether that means minimizing the amount of material used to build a structure, or maximizing the profit earned by a business. Typically, optimizing a quantity involves first writing a function that describes the quantity in question, and then using derivatives to determine the maximum (or minimum, depending on the question) of that function. Consider the following such problems. 1. Minimizing Materials A box is to be designed with a volume of one cubic meter, and it is desired to use the minimum amount of material in its construction. (a) Suppose the box is to have a square base, as depicted in the figure on the right. To minimize the amount of material used to construct the box, find a function for its surface area. Use the fact that the volume of your box is to be one cubic meter to express the area function only in terms of . (b) Minimize the area function from part (a) to give the dimensions of the box that optimizes the amount of material used. Be sure to verify that your answer yields a minimum! One way to do so is to perform a sign analysis, as described in lecture. (You will need to perform such a verification for each of the problems in this assignment.) 2. Maximizing Profit A company is interested (naturally) in determining the number of items to manufacture in order to maximize their profits. To make this determination, we need to find the difference between the company's revenues and costs. In basic economics, a typical cost function is given by a cubic polynomial, i.e., a function of the form () = 3 + 2 + + , where represents the quantity of items a company manufactures (and sells). The number represents the fixed costs, or the cost incurred even if no items are produced, and the terms 3 + 2 + are the variable costs, or the costs that change depending on how many items are produced. These variable costs take into account factors including materials and labor (along with the possibility, say, of getting a better deal the more materials you purchase, or diminishing productivity the more labor is required). (a) Revenue depends on the amount of items that are sold, or the demand for an item. This demand depends on the price of the item, e.g., a higher price induces lower demand. Let's say the demand 1 function for an item is given by = 300 , where is the number of items demanded at a price . 6 Find the revenue function, i.e., a function that gives the amount of income from selling items. (b) Suppose this company has a cost function of () = 0.004 3 1.6 2 + 500 + 16,000 for producing many items. Given your revenue function from part (a), find the profit function, i.e., the amount of money you have after expenses, and maximize this function. MATH-1311 - Brucks - Fall 2016 Optimization Project 3. Sensitivity to Medicine The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form = 2 ( ) , 2 3 where is a positive constant, is the amount of medicine absorbed in the blood, and is a measurement of the body's reaction level (which may be a change in blood pressure, a change in body temperature, etc.). (a) It makes sense to say that the level of reaction is a function of the amount of medicine absorbed . The derivative of with respect to is called the body's sensitivity to the medicine. Find this derivative. (b) Now determine the amount of medicine the body is most sensitive to, i.e., optimize the sensitivity function. (Your answer will involve the unknown constant .) 4. Oil Pipeline An oil refinery is located 1 km north of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south side of the river 6 km east of the refinery. The cost of laying pipe is $400,000/km over land to a point on the north bank and $800,000/km under the river to the tanks. (a) Use the coordinate system suggested in the figure to the right to set up a function for the cost of laying the pipeline, based on the -value of the point . (Note that the point in the figure is just an example; the actual optimization point is yet to be determined.) (b) To minimize the cost function defined in part (a), we need to set the derivative of this function equal to zero. Show that this amounts to solving the equation 4 12 3 + 36 2 16 + 48 = 0. (c) Graph the function = 4 12 3 + 36 2 16 + 48 to choose a whole number initial guess for the solution to the equation in part (b), and use Newton's method to find this solution, correct to three decimal places. Interpret your solution in the context of the original