Rational Function Model Eb48B0

1. **Problem Statement:** Create a real-life situation modeled by a rational function, analyze its properties, graph it, and reflect on the learning. 2. **Real-Life Situation:** Suppose a company produces custom T-shirts. The total fixed cost is 5000 for setup, and the variable cost per shirt is 20. The company sells each shirt for 50. We want to model the profit per shirt as a function of the number of shirts sold. 3. **Formulating the Rational Function:** Let $x$ be the number of shirts sold. Total cost: $$C(x) = 5000 + 20x$$ Total revenue: $$R(x) = 50x$$ Profit: $$P(x) = R(x) - C(x) = 50x - (5000 + 20x) = 30x - 5000$$ To find profit per shirt, divide total profit by $x$ (assuming $x > 0$): $$f(x) = \frac{P(x)}{x} = \frac{30x - 5000}{x} = 30 - \frac{5000}{x}$$ This is a rational function. 4. **Analyzing the Function:** - **Domain:** $x > 0$ (number of shirts sold must be positive) - **Range:** Since $f(x) = 30 - \frac{5000}{x}$, as $x \to \infty$, $f(x) \to 30$ from below; as $x \to 0^+$, $f(x) \to -\infty$. So range is $(-\infty, 30)$. - **x-intercept:** Set $f(x) = 0$: $$0 = 30 - \frac{5000}{x} \Rightarrow \frac{5000}{x} = 30 \Rightarrow x = \frac{5000}{30} = \frac{500}{3} \approx 166.67$$ - **y-intercept:** Not defined since $x=0$ is not in domain. - **Vertical asymptote:** At $x=0$ because denominator zero. - **Horizontal asymptote:** $y=30$ as $x \to \infty$. - **Holes:** None. 5. **Graphing the Function:** The graph has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=30$. The function increases from negative infinity near zero and approaches 30 from below as $x$ increases. 6. **Interpretation:** - The profit per shirt is very low (negative) when few shirts are sold due to fixed costs. - As more shirts are sold, profit per shirt approaches 30. - Selling fewer than about 167 shirts results in negative profit per shirt. 7. **Reflection:** - Rational functions model situations with fixed and variable costs. - They help understand how scale affects profitability. - Challenges include interpreting asymptotes and domain restrictions. - Group collaboration enhances understanding and problem-solving.