Question

Suppose that the cost function for the production of a particular item is given by the function C(x)=2x2−320x+12,020, where x represents the number of items. How many items should be produced to minimize the cost? Explain your answer.

Answer 1

Given function C(x) = $2x^2-320x+12$

Where x represents the number of items.

We need to find number of items that should be produced to minimize the cost.

The given function is a quadratic function. And because coefficent of x^2 is a positive number, therefore parabola open up.

And a quadratic function represents a parabola shape.

The lowest point of the parabola open up is the vertex point.

We already know that we need to find the number of items that should be produced to minimize the cost. So, we need to find the lowest x-coordinate of the parabola.

That is x-coordinate of the vertex point.

The formula for x-coordinate of the parabola is given by.

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

For the given function 2x^2-320x+12, a= 2 and b =-320.

Plugging values of a and b in above formula, we get

x=$\frac{-(-320)}{2(2)}$

$x=\frac{320}{4}$

x=80.

Therefore, 80 items should be produced to minimize the cost.