Problem 76 - Precalculus: Mathematical for Calculus

Question

A biologist is trying to find the optimal salt concentration for the growth of a certain species of mollusk. She begins with a brine solution that has \(4 \mathrm{g} / \mathrm{L}\) of salt and increases the concentration by \(10 \%\) every day. Let \(C_{0}\) denote the initial concentration and \(C_{n}\) the concentration after \(n\) days. (a) Find a recursive definition of \(C_{n}\) (b) Find the salt concentration after 8 days.

Step-by-Step Solution

Answer

(a) \( C_{n} = 1.1C_{n-1} \), (b) \( C_{8} \approx 8.57 \) g/L.

Step 1: Understanding the recursive relationship

The problem states that the salt concentration increases by 10% each day. We can express this mathematically as: \[ C_{n} = C_{n-1} + 0.1C_{n-1} \] This can further be simplified to \[ C_{n} = 1.1C_{n-1} \] where \( C_{0} = 4 \) g/L. This recursive formula allows us to calculate the concentration on any given day based on the previous day's concentration.

Step 2: Express the recursive formula explicitly

To solve part (b), we need to express the recursive formula explicitly so we can calculate \( C_{n} \) directly:\[ C_{n} = 1.1^{n} imes C_{0} \]Substituting \( C_{0} = 4 \) g/L gives:\[ C_{n} = 1.1^{n} imes 4 \]

Step 3: Calculate the concentration after 8 days

Using the explicit formula from Step 2, calculate \( C_{8} \):\[ C_{8} = 1.1^{8} imes 4 \]First, find \( 1.1^{8} \):\[ 1.1^{8} = 2.1436 \] (approximately)Now, multiply by the initial concentration:\[ C_{8} = 2.1436 imes 4 = 8.5736 \] g/L (approximately).

Key Concepts

Geometric Progression

Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. It is a fundamental concept for understanding patterns and structures that recur incrementally, as seen in investment growth or population increase.

In the context of our mollusk study, the salt concentration follows a geometric progression. This is due to the fixed 10% increase each day. We can express this recursive sequence as:

• The first term (\( C_0 \)) is given as 4 g/L.
• The common ratio (\( r \)) is 1.1, representing a 10% increase.
• Thus, each subsequent concentration (\( C_n \)) can be derived by multiplying the previous day's concentration (\( C_{n-1} \)) by 1.1.

This is why the recursive formula becomes \( C_n = 1.1 \times C_{n-1} \).

Exponential Growth

Exponential growth describes a process of increasing quantities at a constant rate per time period, not just by a constant amount. It can be modeled as \( y(t) = y_0 e^{kt} \) in general terms, but specific applications, like our mollusk example, use simpler forms.

With the salt concentration problem, exponential growth comes into play since each day's concentration is not simply adding a fixed amount, but rather increasing by a multiplication factor of 1.1. Therefore, the concentration grows exponentially over time.

• The formula \( C_n = 1.1^n \times C_0 \) captures this growth.
• Here \( C_0 \) is the initial concentration (4 g/L) and \( n \) is the number of days.
• The factor \( 1.1^n \) represents exponential growth, increasing the concentration significantly over multiple days.

Each period (or day) multiplied by 1.1 compounds the concentration further, showing how rapid and powerful exponential growth can be.

Mathematical Modeling

Mathematical modeling involves using mathematical formulas and computations to represent real-world problems and predict changes over time. It is a crucial tool in fields like biology, economics, and engineering.

In the mollusk experiment, mathematical modeling helps the biologist forecast how the salt concentration will change daily. By using the recursive formula \( C_n = 1.1 \times C_{n-1} \) and its explicit version \( C_n = 1.1^n \times C_0 \), the biologist can accurately determine concentrations without daily measurement.
The model

• Starts with defining the initial condition (\( C_0 = 4 \) g/L).
• Uses the consistent percentage increase (10%) to simulate daily conditions.
• Employs recursive and explicit calculations for any given day, illustrating both theoretical and applied science combined.

This method simplifies complex systems, turning iterative processes into straightforward predictions.