Step-by-step explanation:
First, we want to find profit given revenue and cost. Because profit = revenue - cost, we can write the profit function, P(x), as R(x) - C(x) and go from there.
P(x) = R(x) - C(x)
P(x) = (-x²+24x) - ( 10x + 30)
P(x) = -x²+24x - 10x-30
P(x) = -x²+14x-30
In a quadratic function such as this, the graph opens downward (meaning that it goes from up to down) if the leading coefficient is negative and vice versa. Here, since -1 is attached to x², we can say that the graph opens downward. The point where the direction of the slope changes (when it goes from up to down), or the peak of the quadratic, is called the vertex.
The x value of the vertex is given by -b/(2a) in a quadratic of form ax²+bx+c. As a=-1 and b=14 here, we can say that -b/(2a) =
-14/(2*-1) = -14/-2
= 7
Therefore, the x value of the vertex, or the number of containers sold that maximizes profit, is 7.
