A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function C(x)=0.7x^2-462x+96,096. What is the minimum unit cost?

We are given the unit cost function $ C(x) = 0.7x^2 - 462x + 96,096 $, and we want to find the minimum unit cost.

This is a quadratic function of the form $ C(x) = ax^2 + bx + c $. The graph of such a function is a parabola, and since the coefficient of $ x^2 $ (i.e., $ a = 0.7 $) is positive, the parabola opens upwards. Therefore, the function has a minimum at its vertex.

The $ x $-coordinate of the vertex for a quadratic function $ C(x) = ax^2 + bx + c $ can be found using the formula:

\[
x = \frac{-b}{2a}
\]

For the given function:

- $ a = 0.7 $
- $ b = -462 $

Now, calculate the $ x $-coordinate of the vertex:

\[
x = \frac{-(-462)}{2(0.7)} = \frac{462}{1.4} = 330
\]

So, the minimum occurs when $ x = 330 $ cars are produced.

To find the minimum unit cost, substitute $ x = 330 $ into the original function $ C(x) $:

\[
C(330) = 0.7(330)^2 - 462(330) + 96,096
\]

Now, let's calculate it.

\[
C(330) = 0.7(108,900) - 152,460 + 96,096 = 76,230 - 152,460 + 96,096
\]
\[
C(330) = 19,866
\]

Thus, the minimum unit cost is **$19,866** when 330 cars are produced.

Would you like further details or clarification?

Here are 5 related questions:
1. How would increasing production beyond 330 cars affect the unit cost?
2. What happens to the unit cost if fewer than 330 cars are made?
3. How does the vertex formula apply to other quadratic functions?
4. Can you find the number of cars where the unit cost exceeds a certain threshold, like $25,000?
5. How would changes in the coefficients affect the minimum unit cost?

**Tip:** Always check the sign of the quadratic term to know if the parabola opens up (minimum) or down (maximum).

Learn how to find the minimum unit cost of car production using the quadratic function C(x) = 0.7x^2 - 462x + 96,096. Discover the vertex formula for quadratic functions and its application in minimizing costs. Suitable for high school students in Grades 10-12.

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