Question

reyna runs a textile company that manufactures t - shirts. the profit, p, made by the company is modeled by the function $p = s^2 + 9s - 142$, where s is the number of t - shirts sold. how many t - shirts should be sold to earn a profit of more than $2,000?
○ $s < 42$
○ $s > 42$
○ $s > 51$
○ $s < 51$

Explanation:

Step1: Set up the inequality

We want the profit \( p \) to be more than \( \$2000 \), so we set up the inequality \( s^{2}+9s - 142>2000 \).
Subtract 2000 from both sides to get a quadratic inequality in standard form: \( s^{2}+9s-142 - 2000>0 \), which simplifies to \( s^{2}+9s - 2142>0 \).

Step2: Solve the quadratic equation \( s^{2}+9s - 2142 = 0 \)

We can use the quadratic formula \( s=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for a quadratic equation \( ax^{2}+bx + c = 0 \). Here, \( a = 1 \), \( b=9 \), and \( c=- 2142 \).
First, calculate the discriminant \( \Delta=b^{2}-4ac=(9)^{2}-4\times1\times(-2142)=81 + 8568=8649 \).
Then, find the square root of the discriminant: \( \sqrt{8649} = 93 \).
Now, find the roots:
\( s=\frac{-9\pm93}{2\times1} \).
We have two roots:
\( s_1=\frac{-9 + 93}{2}=\frac{84}{2}=42 \) and \( s_2=\frac{-9-93}{2}=\frac{-102}{2}=- 51 \).

Step3: Analyze the quadratic inequality

The quadratic function \( y=s^{2}+9s - 2142 \) is a parabola opening upwards (since \( a = 1>0 \)). The inequality \( s^{2}+9s - 2142>0 \) holds for values of \( s \) less than the smaller root or greater than the larger root.
Since the number of T - shirts sold \( s\) must be a non - negative number, we discard the root \( s=-51 \). So we consider the region where \( s>42 \) (because the parabola opens upwards and we want \( y>0 \), and the roots are \( s = 42 \) and \( s=-51 \)). We can also test a value greater than 42, say \( s = 43 \). Plugging into the profit function \( p=(43)^{2}+9\times43-142=1849+387 - 142=2094>2000 \). Testing a value less than 42, say \( s = 40 \), \( p=(40)^{2}+9\times40-142=1600 + 360-142=1818<2000 \).

Answer:

B. \( s>42 \)