To find the number of T-shirts [tex]\( s \)[/tex] that need to be sold to earn a profit of more than [tex]$2000, we start with the given profit function:
\[ p = s^2 + 9s - 142 \]
We need to determine for which values of \( s \) the profit \( p \) will be greater than $[/tex]2000. Setting up the inequality, we have:
[tex]\[ s^2 + 9s - 142 > 2000 \][/tex]
First, we need to move all terms to one side of the inequality to set it to zero:
[tex]\[ s^2 + 9s - 142 - 2000 > 0 \][/tex]
This simplifies to:
[tex]\[ s^2 + 9s - 2142 > 0 \][/tex]
Now we need to find the roots of the quadratic equation [tex]\( s^2 + 9s - 2142 = 0 \)[/tex] to determine the intervals where this inequality holds.
Using the quadratic formula [tex]\( s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -2142 \)[/tex], we find the solutions:
[tex]\[ s = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 1 \cdot (-2142)}}{2 \cdot 1} \][/tex]
Simplifying inside the square root:
[tex]\[ s = \frac{-9 \pm \sqrt{81 + 8568}}{2} \][/tex]
[tex]\[ s = \frac{-9 \pm \sqrt{8649}}{2} \][/tex]
[tex]\[ s = \frac{-9 \pm 93}{2} \][/tex]
This gives us two solutions for [tex]\( s \)[/tex]:
[tex]\[ s_1 = \frac{-9 + 93}{2} = \frac{84}{2} = 42 \][/tex]
[tex]\[ s_2 = \frac{-9 - 93}{2} = \frac{-102}{2} = -51 \][/tex]
These roots split the number line into intervals that we need to test to see where the inequality [tex]\( s^2 + 9s - 2142 > 0 \)[/tex] holds. The intervals are: [tex]\( (-\infty, -51) \)[/tex], [tex]\( (-51, 42) \)[/tex], and [tex]\( (42, \infty) \)[/tex].
Since the quadratic has a positive leading coefficient (the coefficient of [tex]\( s^2 \)[/tex] is 1), the parabola opens upwards. This means the quadratic expression will be positive outside the roots, i.e., in the intervals [tex]\( (-\infty, -51) \)[/tex] and [tex]\( (42, \infty) \)[/tex].
Therefore, to have a profit greater than [tex]$2000, the number of T-shirts sold \( s \) must be either:
\[ s < -51 \quad \text{or} \quad s > 42 \]
However, since a negative number of T-shirts sold (\( s < -51 \)) does not make sense in this context, the valid solution is:
\[ s > 42 \]
Hence, Reyna must sell more than 42 T-shirts to earn a profit of more than $[/tex]2000.
