khi1la2f1qv

2021-11-29

Use quadratic functions. Suppose that the equation $p\left(x\right)=-2{x}^{2}+280x-1000$, where x represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit?

Michele Tipton

Step 1
We have,
1) $p\left(x\right)=-2{x}^{2}+280x-1000$
This is a quadratic function with and $c=-1000×=-\frac{b}{2a}$
Since $a=-2<0$, which create open downward parabola because 'a' is negative which therefore, creates a maximum at the vertex.
Let us determine the number of terms that should be produced to maximize the cost by find the x-value of the vertex.
We know that, vertex of parabola be:
2) $x=-\frac{b}{2a}$
Substitute the value of $a=-2$ and $b=280$ in equation (2), we get
$⇒x=-\frac{280}{2×\left(-2\right)}$
$⇒x=-\frac{280}{\left(-4\right)}$
$⇒x=\frac{280}{4}$
$⇒x=70$
Therefore, the value of x-coordinates is 70
Hence, the number of items that maximize the profit is 70 items.

Do you have a similar question?