Question

A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function C(x)=1.1x^2-418x+53,580. What is the minimum unit cost?

Answer 1

Answer: 13870

Explanation:

Despite the question has a whole paragraph, it basically asks you when will the function C(x) reach it's minimum point.

I have two ways to solve this problem: Algebra way and Calculus way, and I'll explain both of them.

Algebra way:

There is a formula to determine the x-value of the minimum point of the function: x = (-a)/(2b)

The formula can be used when the function is in the pattern:

y(x) = ax² + bx + c

In this case, a is 1.1, b is -418, and c is 53580.

Therefore, we can find out x-value by (418)/(2 × 1.1) = 190

Now, we just need to substitute x = 190 into the function

and we'll get y = 13870

Calculus way:

I personally like this way because it's faster and easier, but you need to know how to do derivative.

What you need to do is just find out the derivative of the function:

C(x) = 1.1x² - 418x + 53580

C'(x) = 2.2x - 418

Then, we let the derivative C'(x) equal to 0 and find out the x-value:

0 = 2.2x - 418

418 = 2.2x

418/2.2 = x

190 = x

After that, we just need to plug the x into the original function C(x) and find out the y-value