Question

modeling a stores revenue and cost
a stores revenue, $r(x) = -x^2 + 8x$, in thousands of dollars, from selling $x$ thousand units, and its cost, $c(x) = 2x$, in thousands of dollars.
how many units need to be sold to maximize the stores revenue?
4,000 units
6,000 units
16,000 units
12,000 units

Explanation:

Step1: Identify revenue function type

The revenue function $R(x) = -x^2 + 8x$ is a quadratic function in the form $ax^2+bx+c$ where $a=-1$, $b=8$, $c=0$. Since $a<0$, the parabola opens downward, so its vertex is the maximum point.

Step2: Calculate vertex x-coordinate

For a quadratic $ax^2+bx+c$, the x-coordinate of the vertex is $x = -\frac{b}{2a}$.
Substitute $a=-1$, $b=8$:
$$x = -\frac{8}{2(-1)} = \frac{8}{2} = 4$$

Step3: Convert to actual units

$x$ represents thousands of units, so $4 \times 1000 = 4000$ units.

Answer:

4,000 units