help needed with math

Question

A medical equipment industry manufactures X-ray machines. The unit cost c

(the cost in dollars to make each X-ray machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function 

c(x)=0.6x^2-108x+19,222. What is the minimum unit cost?

Answer 1

Hi Mariah,

your function is a parabola graph or a square function and you are looking for the vertex as it is the lowest possible value for a parabola that is open on top (which you can see by the fact that the number before your x² is positive.

There are multiple ways to find this solution. The easiest would be to use the formula to determine the vertex:

In order to understand the formula lets look at your function first.

All square functions look like this:

y= ax² + bx +c , with a,b and c being variables / placeholders for real numbers.

Your function reads: y= 0.6 x² - 108 x + 19.222

so the value for a= 0.6 b=108 and c=19.222

The x coordinate of the vertex can be calculated with this formula:

X_vertex= -b / (2*a)

The y coordinate can be calculated with this formula:

y= c - (b²) / (4*a)

So lets calculate your coordinates

x = -(-108)/ (2*0.6)= 90

y = 19,222 - (-108)² /(4* 0.6) = 14,362

So according to the numbers your lowest unit price y = $ 14,362

for x = 90 machines being produced.

Feel free to shot me a message if you have any further questions about the answer, vertex, square functions or the topic in general, or if you want to discuss the alternative solutions.