To find the minimum unit cost for the given function $c(x) = 1.2x^2 - 504x + 71.791$, we need to find the vertex of the quadratic function. The vertex of a quadratic function $ax^2 + bx + c$ occurs at the $x$-coordinate given by:

$x = \frac{-b}{2a}$

For the function $c(x) = 1.2x^2 - 504x + 71.791$, the coefficients are:

• $a = 1.2$
• $b = -504$

Substituting these values into the formula for the $x$-coordinate of the vertex:

$x = \frac{-(-504)}{2(1.2)} = \frac{504}{2.4} = 210$

Thus, the number of machines that minimizes the unit cost is $x = 210$.

Next, we substitute $x = 210$ into the cost function to find the minimum unit cost:

$c(210) = 1.2(210)^2 - 504(210) + 71.791$

Now, let's calculate:

$c(210) = 1.2(44100) - 504(210) + 71.791$ $c(210) = 52920 - 105840 + 71.791$ $c(210) = -52920 + 71.791 = -52848.209$

Thus, the minimum unit cost is $-52848.209.

Would you like more details or have any questions? Here are 5 related questions:

1. How does the number of machines made affect the unit cost?
2. What would happen to the unit cost if the company made fewer or more than 210 machines?
3. How would you find the maximum value of a quadratic function?
4. Why does the vertex of a quadratic function represent the minimum or maximum value?
5. What is the effect of changing the coefficients $a$, $b$, and $c$ in a quadratic function?

Tip: When dealing with quadratic functions, the vertex formula is a reliable way to find the minimum or maximum value without needing to graph the function.