To find out how many T-shirts should be sold to achieve a profit of more than \[tex]$2000, we start with the given profit function:
\[ p = s^2 + 9s - 142 \]
where \( p \) is the profit and \( s \) is the number of T-shirts sold. We need to determine the values of \( s \) for which the profit \( p \) exceeds \$[/tex]2000.
1. Set up the inequality:
[tex]\[ s^2 + 9s - 142 > 2000 \][/tex]
2. Convert the inequality into an equation:
[tex]\[ s^2 + 9s - 142 = 2000 \][/tex]
3. Move all terms to one side to form a standard quadratic equation:
[tex]\[ s^2 + 9s - 2142 = 0 \][/tex]
4. Solve the quadratic equation [tex]\( s^2 + 9s - 2142 = 0 \)[/tex] to find the critical points.
By solving this quadratic equation, we determine the roots as follows:
[tex]\[ s = -51 \][/tex]
[tex]\[ s = 42 \][/tex]
These roots represent the points where the profit exactly equals \[tex]$2000. To find when the profit is greater than \$[/tex]2000, we need to test the intervals determined by these roots.
5. Analyze the intervals defined by the roots:
The quadratic function [tex]\( s^2 + 9s - 2142 \)[/tex] is a parabola opening upwards (the coefficient of [tex]\( s^2 \)[/tex] is positive). This means that the profit will be greater than \[tex]$2000 outside the interval \([-51, 42]\).
6. Determine the intervals where the profit exceeds \$[/tex]2000:
The function [tex]\( s^2 + 9s - 2142 > 2000 \)[/tex] will hold true for:
[tex]\[ s < -51 \quad \text{or} \quad s > 42 \][/tex]
However, the number of T-shirts [tex]\( s \)[/tex] sold cannot be negative. Therefore, the only practically relevant solution is:
[tex]\[ s > 42 \][/tex]
Thus, Reyna should sell more than 42 T-shirts to achieve a profit of more than \$2000.
Answer:
[tex]\[ s > 42 \][/tex]
