Problem 78 - Algebra and Trigonometry

Question

Concentration of a Solution 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 \(C_{m}\) (b) Find the salt concentration after 8 days.

Step-by-Step Solution

Answer

(a) \( C_n = 1.10 \times C_{n-1} \), \( C_0 = 4 \). (b) 8.57 g/L after 8 days.

Step 1: Understand the Problem Parameters

To solve the problem, you need to determine how a given salt concentration changes day by day. Initially, the concentration is noted as \( C_0 = 4 \, \text{g/L} \). Each subsequent day sees an increase in this concentration by 10\%. The problem asks for a recursive formula to express this sequential increase, and to determine the concentration after 8 days.

Step 2: Establish the Recursive Formula

A recursive formula defines each term based on the previous term. Here, the concentration each day, \( C_n \), is derived by increasing the previous day's concentration, \( C_{n-1} \), by 10\%. This can be expressed as:\[ C_n = C_{n-1} + 0.10 \times C_{n-1} = 1.10 \times C_{n-1} \]Thus, the recursive definition is \( C_n = 1.10 \times C_{n-1} \). The base case is \( C_0 = 4 \, \text{g/L} \).

Step 3: Iteratively Apply the Recursive Formula

Using the formula \( C_n = 1.10 \times C_{n-1} \), calculate the concentration day by day. Start with \( C_0 = 4 \):- \( C_1 = 1.10 \times 4 = 4.4 \, \text{g/L} \)- \( C_2 = 1.10 \times 4.4 = 4.84 \, \text{g/L} \)- \( C_3 = 1.10 \times 4.84 = 5.324 \, \text{g/L} \) - Continue applying this pattern until you reach \( C_8 \).

Step 4: Calculate the Concentration After 8 Days

Continuing the process from step 3, calculate sequential concentrations:- \( C_4 = 1.10 \times 5.324 = 5.8564 \, \text{g/L} \)- \( C_5 = 1.10 \times 5.8564 = 6.44204 \, \text{g/L} \)- \( C_6 = 1.10 \times 6.44204 = 7.086244 \, \text{g/L} \)- \( C_7 = 1.10 \times 7.086244 = 7.7948684 \, \text{g/L} \)- \( C_8 = 1.10 \times 7.7948684 = 8.57435524 \, \text{g/L} \)Therefore, the salt concentration after 8 days is approximately \( 8.57 \, \text{g/L} \).

Key Concepts

Salt Concentration

Salt concentration refers to the amount of salt dissolved in a solution, often measured in grams per liter (g/L). In the context of aquatic organisms and experiments like the one described, salt concentration is a crucial parameter. It can influence biological processes such as osmotic pressure, which affects cells, so the right concentration is vital.
In this exercise, the salt concentration begins at 4 g/L. The task is to understand how this concentration changes over time, with a consistent daily percentage increase, to ensure optimality for mollusk growth.
Tracking these changes can help determine how organisms react to varying environments and optimize growth conditions in controlled settings.

Growth of Species

Growth of species, particularly in environments regulated by salt concentration, is a nuanced process. Salinity plays a key role in physiological conditions affecting growth rates.
When studying species growth, like the biologist in this exercise, altering the environment iteratively (as we do by changing salt concentration) provides insight into growth patterns.

• Species like mollusks require specific conditions for optimal growth. Changes in these conditions can begin at a cellular level.
• Understanding how a species adapts to increasing salt concentrations helps biologists control and manage ecological and aquacultural settings better.

This exploration emphasizes ecological balance and dietary needs for various aquatic species.

Percent Increase

A percent increase is used to describe the rise in a quantity by a specific percentage. In this exercise, the salt concentration sees a 10% increase each day. This means that every new day's concentration is 10% more than the previous day's amount.
To calculate a percent increase:

• Identify the original quantity (e.g., 4 g/L).
• Determine the percentage increase (e.g., 10%).
• Multiply the original quantity by the percentage increase and add the result to the original.
• Alternatively, multiply by a factor of 1 plus the increase rate (i.e. 1.10 for a 10% increase).

This method effectively tracks growth rates and is essential for managing dynamic conditions like those in biological experiments.

Recursive Formula

A recursive formula is a mathematical way to express terms in a sequence based on preceding terms. It’s particularly useful to describe gradual processes or repeated actions, such as percent increases over time.
In our exercise, the recursive formula allows us to calculate daily salt concentrations without directly computing each day independently, capturing how today’s concentration depends on yesterday’s. It is outlined as:

• Base case: Start with an initial value, here it is \( C_0 = 4 \, \text{g/L} \).
• Recurrence relation: \( C_n = 1.10 \times C_{n-1} \).

This setup gives each term in a sequence context from its past, making it powerful for understanding trends and predicting future outcomes in incremental systems like biological concentration studies.