Expert: Scott A Wilson - 4/16/2011

Question
Hey Scott I really could use your help again! USA manufacturing has determined that when x units are produced, the average cost per product is given by: C(x)=0.2x^2-1.3x+3.4. Where C(x) is in hundreds of dollars. What is the minimum cost per product and ow many products should be produced to achieve that minimum?

Thanks for all your help!

Answer
To find the minimum, first the total revenue must be found.
That would be the number of products times the number of products.
In other words, the total price P(x) would be x*C(x).
This makes P(x) = 0.2x^3 - 1.3x^2 + 3.4x.

To find the minimum, take the derivative of P(x), set it to 0, and use the binomial theorem.

We can see that P'(x) = 0.6x^2 - 2.6x + 3.4.
The solutions are given by x = (-b +/- √(b^2 - 4ac))/(2a) where a = 0.6, b = -2.6, and c = 3.4.
It might be useful to known b^2 = 2.6^2 = 6.76 and 4ac = 8.16.

The value of x is the number of products and 100*C(x) is the cost.