1.

\fFind the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 92 meters Find two positive numbers whose sum is 190 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) First Second Number, X Number Product, P 50 190 - 50 50(190 - 50) = 7,000 50 190 - 60 60(190 - 60) = 7,800 70 190 - 70 70(190 - 70) = 80 190 - 80 80(190 - 80) = 90 190 - 90 90(190 - 90) = 100 190 - 100 100(190 - 100) = (b) Write the product P as a function of x. P ( X ) = (c) Use calculus to find the critical number of the function in part (b). Then find the two numbers. (Enter your answers as a comma-separated list.) (d) Use a graphing utility to graph the function in part (b) and verify the solution from the graph. 10000 8000 9000 150 8000 6000 8950 100 6000 4000 4000 2000 8900 50 2000 20 40 60 80 100 120 140 160 180 200 8850 O 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 -2000 O 0 20 40 60 80 100 120 140 160 180 200 The solution appears to be at x =