Problem 58 - Contemporary Precalculus

Question

In a laboratory culture, the number \(N(d)\) of bacteria (in thousands) at temperature \(d\) degrees Celsius is given by the function $$ N(d)=\frac{-90}{d+1}+20 \quad(4 \leq d \leq 32) $$ The temperature \(D(t)\) at time \(t\) hours is given by the function \(D(t)=2 t+4 \quad(0 \leq t \leq 14)\) (a) What does the composite function \(N \circ D\) represent? (b) How many bacteria are in the culture after 4 hours? After 10 hours?

Step-by-Step Solution

Answer

Question: Given the functions \(N(d) = \frac{-90}{d + 1} + 20\) and \(D(t) = 2t + 4\), find the composite function \(N \circ D\) and determine what it represents. Then, find how many bacteria (in thousands) are in the culture after 4 and 10 hours. Answer: The composite function \(N \circ D(t) = \frac{-90}{2t + 5} + 20\) represents the number of bacteria in thousands present in the culture at a given time \(t\) in hours, taking into account the temperature change over time. There are approximately 13.077 thousand bacteria in the culture after 4 hours and 16.4 thousand bacteria after 10 hours.

Step 1: Finding the composite function \(N \circ D\)

First, let's find the composite function, \(N \circ D\). This can be found by substituting the function for \(D(t)\) into the function for \(N(d)\). So we have: $$N \circ D(t) = N(D(t)) = N(2t + 4)$$ Now replace \(d\) with \((2t + 4)\) in the \(N(d)\) equation: $$N(2t + 4) = \frac{-90}{(2t + 4) + 1} + 20$$

Step 2: Simplifying the composite function

Now, let's simplify the composite function \(N \circ D(t)\): $$N \circ D(t) = \frac{-90}{2t + 5} + 20$$ The composite function \(N \circ D(t)\) represents the number of bacteria in thousands present in the culture at a given time \(t\) in hours, taking into account the temperature change over time.

Step 3: Finding the number of bacteria at 4 hours

Now that we have our composite function, we can find the number of bacteria present after 4 hours by substituting \(t = 4\) in the function: $$N \circ D(4) = \frac{-90}{2(4) + 5} + 20 = \frac{-90}{13} + 20 \approx 13.077$$ There are approximately 13.077 thousand bacteria in the culture after 4 hours.

Step 4: Finding the number of bacteria at 10 hours

Similarly, we can find the number of bacteria present after 10 hours by substituting \(t = 10\) in the function: $$N \circ D(10) = \frac{-90}{2(10) + 5} + 20 = \frac{-90}{25} + 20 = 16.4$$ There are 16.4 thousand bacteria in the culture after 10 hours.

Key Concepts

Composite Functions

Composite functions are the result of applying one function to the results of another function. Think of it as a sequence where one process follows another. In precalculus, understanding composite functions is paramount for analyzing situations where multiple actions or transformations are applied in succession. For instance, if you have a function that describes temperature over time, and another that models the growth rate of bacteria based on temperature, combining these functions shows the relationship between time and bacterial growth directly.

To build a composite function like in our exercise with the functions for bacteria number and temperature, you plug in the definition of one function into the other. The notation we use is \( N \circ D \) to indicate that we first apply function \( D\) and then function \(N\) to whatever input we have. This becomes crucial when trying to understand complex scenarios where one variable directly depends on another that is itself changing.

Functions in Precalculus

In the realm of precalculus, functions are the bread and butter for modeling relationships between two variables. They are mathematical tools that take an input and provide an output, connecting different quantities in a consistent and predictable manner. The power of functions in precalculus lies in their utility for abstraction. Instead of dealing with different numbers individually, a function can describe whole sets of possible scenarios with a single expression.

Functions not only provide clarity through this abstraction but also enable us to apply various mathematical techniques such as transformations, combinations, and analysis to reveal deeper insights. These details can lead to better comprehension in various real-world applications from science to economics.

Modeling with Functions

Modeling with functions involves creating a function or a set of functions that describe real-world phenomena. In our exercise, we set up functions that model the relationship between bacteria growth and temperature. This kind of modeling is extremely valuable for making informed predictions about the behavior of complex systems. It is how mathematicians, scientists, and engineers translate real-life problems into a language that can be understood, manipulated, and solved with mathematical tools.

Crucially, the success of a model depends on how well the function represents the actual conditions. In the given problem, we assume that the bacterial growth solely depends on the temperature, which is a simplification, but it's a starting point for creating a model that gets us closer to understanding the dynamics at play. With a sound model, you can adjust inputs to see how the system responds, aiding in decision-making.

Real-World Applications of Functions

The real-world applications of functions are nearly endless. By translating aspects of reality into mathematical language, we can use functions to study everything from the trajectory of a satellite to the swing of a pendulum. In biology, functions can model population dynamics, like in the exercise where we related temperature to the growth of bacteria.

In economics, they can represent supply and demand curves. In physics, they're crucial to explain motion, force, and energy. And that's just the beginning; health sciences, social sciences, engineering, and more lean heavily on the power of functions to both describe and predict outcomes. Good modeling can save time, lives, and resources by allowing us to simulate and understand complex behaviors without needing to physically construct or witness those scenarios.