Question

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^2-420x+81,610. How many cars must be made to minimize the unit cost? Do not round your answer.

Answer 1

Answer:

300 cars must be made to minimize the unit cost

Step-by-step explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

$f(x) = ax^{2} + bx + c$

It's vertex is the point $(x_{v}, y_{v})$

In which

$x_{v} = -\frac{b}{2a}$

$y_{v} = -\frac{\Delta}{4a}$

Where

$\Delta = b^2-4ac$

If a>0, the vertex is a minimum point, that is, the minimum value happens at $x_{v}$, and it's value is $y_{v}$.

The cost of producing x cars is given by:

$C(x) = 0.7x^2 - 420x + 81610$

So a quadratic equation with $a = 0.7, b = -420, c = 81610$

How many cars must be made to minimize the unit cost?

This is the xvalue of the vertex. So

$x_v = -\frac{b}{2a} = -\frac{-420}{2*0.7} = \frac{420}{1.4} = 300$

300 cars must be made to minimize the unit cost