Problem 78 - Precalculus Mathematics for Calculus

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 \(C_{n}\) (b) Find the salt concentration after 8 days.

Step-by-Step Solution

Answer

Recursive formula: \(C_{n+1} = C_{n} \times 1.10\); Concentration after 8 days: \(8.57\) g/L.

Step 1: Identify the Initial Concentration

The initial concentration of the salt solution is given as \(C_{0} = 4\) g/L. This serves as the starting point for determining the concentration on subsequent days.

Step 2: Determine the Daily Increase in Concentration

Each day, the concentration increases by 10%. Therefore, the concentration after a day can be calculated as the previous day's concentration multiplied by 1.10. This can be expressed as a formula: \(C_{n+1} = C_{n} \times 1.10\).

Step 3: Write the Recursive Definition

Using the information from Step 2, the recursive definition for the concentration \(C_{n}\) after \(n\) days is given by:- \(C_{0} = 4\) (initial concentration)- \(C_{n+1} = C_{n} \times 1.10\) (recursive formula)

Step 4: Calculate the Concentration After 8 Days

Using the recursive formula, calculate the concentration for each day up to 8 days:- \(C_{1} = 4 \times 1.10 = 4.4 \)- \(C_{2} = 4.4 \times 1.10 = 4.84 \)- \(C_{3} = 4.84 \times 1.10 = 5.324 \)- \(C_{4} = 5.324 \times 1.10 = 5.8564 \)- \(C_{5} = 5.8564 \times 1.10 = 6.44204 \)- \(C_{6} = 6.44204 \times 1.10 = 7.086244 \)- \(C_{7} = 7.086244 \times 1.10 = 7.7948684 \)- \(C_{8} = 7.7948684 \times 1.10 = 8.57435524 \)Thus, the concentration after 8 days is approximately \(8.57\) g/L.

Key Concepts

Recursive Formula

A recursive formula is a way of defining a sequence where each term is determined based on its predecessor. It's like stepping stones in a river, where each step relies on the previous one.
In the given exercise, the recursive definition helps us know the salt concentration for each day. We start with the initial concentration, which is 4 g/L. Then, the recursive formula conveys how the concentration changes daily. This change is described as:

• Starting point: \(C_0 = 4\) g/L
• Recurrence relation: \(C_{n+1} = C_n \times 1.10\)

Here, \(C_{n+1}\) presents the concentration on the next day, while \(C_n\) is the current day's concentration. This format is particularly useful because it provides a systematic approach to calculate concentrations for any number of days with just the starting value and the relation.

Exponential Growth

Exponential growth occurs when a quantity increases by the same proportion over equal time intervals. This pattern is evident with the mollusk salt concentration. Exponential growth is characterized by its rapid increase, initially slow but accelerating over time. Let's break it down:

• The concentration increases by 10% daily, meaning each day's concentration is 1.10 times the previous day's.
• Despite beginning at a modest 4 g/L, the concentration grows faster as days go by due to the compound effect of multiplying by 1.10 repeatedly.

This phenomenon is widespread across different fields, from finance to population studies, because many real-world processes exhibit exponential patterns. Understanding this can help predict future scenarios based on past data, like estimating concentrations after additional days.

Concentration Calculation

Understanding the concentration change over days can enable predicting future conditions. In our exercise, calculating the salt concentration after 8 days was straightforward due to the recursive formula.To calculate the concentration for a particular day, you multiply the concentration from the previous day by 1.10. This process is repeated until reaching 8 days:

• Day 1: \(C_1 = 4 \times 1.10 = 4.4\)
• Day 2: \(C_2 = 4.4 \times 1.10 = 4.84\)
• Continue multiplying by 1.10 until Day 8: \(C_8 = 8.57\) g/L

Such calculations are essential across various scientific disciplines. In biology, they can assist with determining the impact of changing environments on species growth. Understanding how to employ these calculations aids in crafting well-informed predictions and decisions.