4.3: Rational Functions--Homework

Rational Functions Homework Problems

1. Economic Modeling: A startup tech company has initial fixed costs of $12,000 for equipment and software. It costs the company $15 to produce each smart-home sensor. Let \(x\) represent the number of sensors produced.
   1. Find a formula for the total cost function, \(C(x)\).
   2. Find a formula for the average cost per sensor, \(A(x) = \frac{C(x)}{x}\).
   3. Find \(\lim_{x \to \infty} A(x)\). Explain what this value represents in the context of the company’s long-term production.
   4. How many sensors must be produced to bring the average cost down to $25 per sensor?
2. Chemical Mixtures: A 50-liter tank initially contains 5 kg of salt dissolved in water. Pure water is being added to the tank at a rate of \(t\) liters. The concentration of salt, \(C\), in kg/L, is given by the function: \[C(t) = \frac{5}{50 + t}\]
   1. What is the initial concentration of salt when \(t = 0\)?
   2. Evaluate \(C(50)\) and explain its meaning.
   3. Find the horizontal asymptote of \(C(t)\). In the context of the tank, why does this asymptote make sense?
   4. Find the inverse function \(C^{-1}(y)\) and explain what it calculates.
3. Asymptotic Behavior: For each of the following rational functions, identify the leading terms of the numerator and denominator, then determine the horizontal asymptote if one exists:
   1. \(f(x) = \dfrac{4x^2 - 3x + 1}{2x^2 + 9}\)
   2. \(g(x) = \dfrac{10x + 5}{x^2 - 4}\)
   3. \(h(x) = \dfrac{x^3 - 1}{x^2 + 1}\)
   4. For part (a), evaluate \(\lim_{x \to \infty} f(x)\). How does this limit relate to the horizontal asymptote?

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