The product of two positive, real numbers is 80. What is the minimum sum of one of the numbers plus the square of the other? Solution: Step 1: Let a: and y be the two variables and s the sum we are interested in. 34+)! Xy Step 2: We have s : "3" c and thy ': 80. xAz-i-y _ x"2+y"2 X+y ' 80/): Step 3: Solving for y in the second equation, we have 3; :' 80): '. So the fundamental equation becomes .9 : 80x A an 80x x"- 2 +(BCij: 31+ [ELDER] 8 0 + X Step 4: The domain of s is all positive, real numbers. 40 _2):-[aD,rx] 80 _ Step 5: Taking the derivative, we have 3: : 31:3):"23, ' The critical value in the domain is m : :DAH-B) : . . . Turanmz) . . . . . . . On the domain, a Is decreasrng before the critical value and Increasmg after, so there Is a minimum at the critical value. Step 6: The minimum sum is s : 402/3 + 80 - 401/3. EjFind the minimum sum of two positive numbers whose product is 900. u a. 330 C b. 33 C c. 50 C d. 930 e. 60 C) 2A power line is to be run to an offshore facility in the manner described in Example 4.3.3. The offshore facility is 8 km along the shoreline from the power plant and 3 km at sea from there. It costs $40 ODD/km to lay a power line underground and $70 000 to lay a power line underwater. How much of the power line should be run underground, in km, to minimize the total costs? We will approximate V8.5 using differentials. 8.5 First, let y X 8 x^ (1/3) 9^(1/3) X^ (1/2) For the close, known value, we will use 8^(1/3) 2. 9^(1/2) 3 1 For the differential, we have dy X"(-2/3) dx. Since x - 8.5 and da = 0.01 - we have (1/3)x^(-2/3) N dy = 1/12. 0.5 x^ (1/3) 8 (1/3)x 0.1 2 Therefore, V8.5 ~ 1/12 2+1/12