To determine the number of items the company needs to sell each week to maximize its profit and what the maximum profit will be, we need to analyze the given quadratic profit function:
[tex]\[ P(x) = -0.5x^2 + 40x - 300 \][/tex]
This function is a parabola that opens downwards (since the coefficient of [tex]\( x^2 \)[/tex] is negative), and the maximum profit will occur at its vertex.
1. Finding the Vertex:
For a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the vertex, which gives the maximum (or minimum) value, is found at:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
In our profit function, [tex]\( a = -0.5 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( c = -300 \)[/tex].
Substituting these values into the vertex formula:
[tex]\[
x = -\frac{40}{2 \cdot -0.5} = -\frac{40}{-1} = 40
\][/tex]
Therefore, the company should sell [tex]\( 40 \)[/tex] items each week to maximize its profit.
2. Calculating the Maximum Profit:
To find the maximum profit, we substitute [tex]\( x = 40 \)[/tex] back into the profit function [tex]\( P(x) \)[/tex]:
[tex]\[
P(40) = -0.5(40)^2 + 40(40) - 300
\][/tex]
Calculating inside the parentheses and performing the operations step-by-step:
[tex]\[
P(40) = -0.5(1600) + 1600 - 300
\][/tex]
[tex]\[
P(40) = -800 + 1600 - 300
\][/tex]
[tex]\[
P(40) = 800 - 300
\][/tex]
[tex]\[
P(40) = 500
\][/tex]
Therefore, the maximum profit is [tex]\( \$500 \)[/tex].
In summary, the company should sell 40 items each week to achieve a maximum profit of \$500.
