Can anyone figure out this easy question (its hard really >.<)?

Question

The annual cost of production of a ton of a chemical is given by the following formula, where x is the number of tons per year produced.

Cost per ton = 1,000,000 / x + 100 + x

can you find the value of x for which the cost per ton is a minimum, and the cost per ton for this value of x.

could you show working out so i can learn how to do similar questions. thanks

Ahh thank you guys :)

Answer 1

This is a calculus optimization problem. If you have no calculus experience, this won't make sense, but here goes:

Set f(x) = 1,000,000/x + 100 + x then take the derivative.

f'(x) = -1,000,000/x^2 + 1

Now, set the derivative equal to zero and solve for x.

x = 1000

Check values of x less than and greater than 1000 by plugging them into the derivative to confirm that this is a minimum.

Now, plug 1000 into the original function to get the cost per ton for that value of x.

The final answers are: When x is 1000, the cost per ton is a minimum and the cost per ton is 2100.

Hope this helps!

Answer 2

C = 1,000,000/x + 100 + x
C' = -1,000, 000x^-2 + 1
0 = -1,000,000x^-2 + 1
0 = -1,000,000 + x^2
1,000,000 = x^2
1,000 = x

Cost per ton at that point = 1,000 + 100 + 1,000 = 2,100

The derivative is used to find a minimum, maximum of a curve. You could also graph it on a graphing calculator and see where the bottom of the curve is. Or you can plug numbers into an Excel spreadsheet.