The cost C in dollars of operating a certain concrete machin

The cost C in dollars of operating a certain concrete machine is related to the number of minutes n the machine is run by the function: C(n) = 2.2n^2 - 66n + 655 For what number of minutes is the cost of running the machine a minimum? minutes (Type a whole number) What is the minimum cost? $ (Type an integer or a decimal)

Solution

We have C(n) = 2.2n² - 66n + 655. The cost C is minimum when dC/dn = 0 and d²C/dn² is positive.

Now, dC/dn = 2.2 *2n - 66 = 4.4n - 66 so that if dC/dn = 0, then 4.4n - 66 = 0 or, 4.4n = 66 so that n = 66/4.4 = 15 minutes. Also, d²C/dn² = 4.4 which is positive. Therefore, the cost C(n) is minimum when the Cocrete mixer is run for 15 minutes.

The minimum cost is C(15) = 2.2(15)² -66(15) + 655 = 2.2(225) - 990 + 655 = 495 - 990 + 655 = $ 160.