Algebra question

Question

An aircraft factory manufactures airplane engines. The unit cost c (the cost in dollars to make each airplane engine) depends on the number of engines made. If x engines are made, then the unit cost is given by the function c(x)=1.1x^2-484x+65259 . What is the minimum unit cost?

Answer 1

Could also take the first derivative and set to 0 to find minimum.

c'(x) = 2.2x - 484 = 0

x = 220, etc.

Answer 2

First, we have to recognize that we want to solve for the minimum c(x) value in your equation:

c(x) = 1.1x² – 484x + 65259

Let's first find the x value of this minimum. We can do this by solving the following:

x = -b / 2a

where a and b come from the equation in the form ax² + bx + c

Note that if this parabola opened downward instead of upward—in which case the a value would be negative—this formula would be used to find the x coordinate of the maximum value.

We can see that in this example:

a = 1.1

b = -484

c = 65259

Plugging those numbers in:

x = -b / 2a

= -(-484) / 2(1.1)

= 484 / 2.2

= 220

From there, we just have to solve the original equation to get the c(x) value that corresponds with x = 220:

c(x) = 1.1x² – 484x + 65259

c(220) = 1.1(220)² – 484(220) + 65259

= 12019

Therefore, the minimum unit cost is $12,019.