The problem is a good one for groups of students for the difficulty in the second problem is usually overcome by groups more easily than individuals, for the discussion which ensues in groups does not get stymied as one individual might. Expectations We would expect students to understand and formulate problem 1 quite routinely. As to problem 2 we expect students to appear to be frustrated at first, especially when these problems are juxtaposed and they thus see that the simple strategy which worked in problem 1 will not work in problem 2 as they cannot get their hands on the objective function so readily. But we expect the students to examine the possible range of values of the independent variable, to test out some values per the suggestion in the statement of the problem, and then to either make list, trial and error, or fit function to several data points and optimize either by inspection of table or plot of data, or fit a function to the data and proceed to optimize based on that fitted function. Future payoffs Students will see that even though they cannot formulate a function in closed form they may be able to optimize it.
Extensions Use different shapes in different regions.
Be sure you make clear your strategy for solving this problem, even if you do not complete the analysis. Thus we see the maximum area is 0.561096 square units. 0.561096 Solutions to Problem 2. From the combination of the data and the interpolating polynomial graphs we see the polynomial is a good estimate of the actual data in the maximum region which interests us especially. We could have gone after a cubic polynomial just as easily.