Evaluate the integral S ac7 (208 - 4 15 da by making the appropriate substitution: u [answer1] J ac7 (208 - 4) 15 dx = + C [answer2]Find two positive numbers, and , such that x + y = 14 and they minimize ac2 + yr . a = y = Detailed Solution We need to minimize S(x) = x + y with the constraint a + y = 14. Solve for y in the 2nd equation y = 14 - x And substitute into the minimizing function and simplify: S(x) = 2 + (14-2)2 S(x) = 12 + 14' - 2 . 14 . x + 22 S(x) = 2x2 - 28x + 196 Find the critical values by finding the derivative and setting it equal to 0. S'(x) = 4x - 28 0 = 4x - 28 T = 7You can run at a speed of 6 mph and swim at a speed of 3 mph and are located on the shore, 4 miles east of an island that is 1 mile north of the shoreline. How far should you run west to minimize the time needed to reach the island? Island 1 l+l You Write the equation that needs minimizing. Use x as the miles west you will run before swimming to the island. Detailed Solution