Problem 84 - Precalculus: Mathematics 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 let \(C_{n}\) be 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) \(C_n = 1.1^n \times 4\); (b) 8.5744 g/L after 8 days.

Step 1: Understand the Problem

The biologist is increasing the salt concentration of a brine solution every day by 10%. We start with an initial concentration of 4 g/L of salt (denoted as \(C_0\)). We need to derive a recursive formula for the concentration after \(n\) days (\(C_n\)), and calculate this concentration after 8 days.

Step 2: Define the Recursive Formula for \(C_n\)

The concentration is increased by 10% every day, so each new concentration is 110% of the previous day's concentration. The recursive definition can be expressed as:\[ C_{n} = C_{n-1} + 0.1C_{n-1} = 1.1C_{n-1} \]Thus, \(C_n = 1.1^n C_0\) and since \(C_0 = 4\), the recursive formula is:\[ C_n = 1.1^n imes 4 \]

Step 3: Calculate the Salt Concentration After 8 Days

Use the recursive formula \(C_n = 1.1^n imes 4\) with \(n = 8\):\[ C_8 = 1.1^8 imes 4 \]Calculate \(1.1^8\) first:\(1.1^8 \approx 2.1436\).Then, compute \(C_8\):\[ C_8 \approx 2.1436 imes 4 = 8.5744 \]Thus, after 8 days, the concentration is approximately 8.5744 g/L.

Key Concepts

Salt Concentration

Salt concentration in a solution refers to the amount of salt present per volume of liquid. It's denoted as grams of salt per liter (g/L) in this scenario. For the growing mollusks, maintaining an appropriate salt concentration is crucial. The initial concentration, termed as \(C_0\), indicates the starting point of the experiment, which is 4 g/L in this case. Each subsequent day, the salt concentration is intended to be increased by a certain percentage to find the optimum levels for mollusk growth.

To express this change mathematically, a recursive sequence is utilized. It describes how each new day's concentration (\(C_n\)) is derived from the previous day's concentration (\(C_{n-1}\)). The rule in this context is that the new concentration is 110% of the old concentration, leading to the formula \(C_n = 1.1 \times C_{n-1}\). This shows a steady increase, driven by multiplying the concentration of the previous day by 1.1, effectively adding 10% more salt each day. This pattern continues over multiple days to reach the desired concentration levels.

Growth Rate

Understanding the growth rate of a factor such as salt concentration is crucial in biological experiments. In this scenario, the growth rate signifies how much the concentration increases daily. The biologist chose to increase the salt concentration by 10% daily to determine the impact on mollusk growth. This strategy involves using a consistent multiplier, 1.1, which represents the original concentration plus the added 10% for growth over time.

Growth rate is important because it directly affects the conditions provided to organisms in experiments. Calculating the daily increase helps researchers predict future conditions accurately. Using recursive sequences, the formula \(C_n = 1.1^n \times 4\) is established. Here, \(n\) denotes the number of days, and the initial concentration is multiplied by \(1.1^n\) to reflect the compounding effect of a constant growth rate over time. For instance, after 8 days, the salt concentration grows significantly, calculated as \(1.1^8 \approx 2.1436\) times its original value, influencing the experimental environment for the mollusks.

Mollusk Cultivation

Mollusk cultivation involves creating the optimal environment for mollusks to grow and thrive. Salt concentration plays a key role in this process, as different species of mollusks may require varying levels of salinity for optimal growth. The experiment outlined seeks to find how the mollusks respond to gradual increases in salt concentration over time, which may mimic their natural habitat changes.

To successfully cultivate mollusks, biologists must monitor environmental factors such as water temperature, salinity, and pH levels. Ensuring the salt concentration changes in a controlled manner allows the mollusks to adapt and potentially enhance their growth rate. This experiment's incremental approach, increasing salt by 10% daily, may provide insights into the mollusks' tolerance and preference, guiding further cultivation efforts. Adjustments in such variables might help optimize conditions for their development and increase the yield of mollusk farming over time.