Question

[T] Suppose you start with one liter of vinegar and repeatedly remove \(0.1 \mathrm{~L}\), replace with water, mix, and repeat. a. Find a formula for the concentration after \(n\) steps. b. After how many steps does the mixture contain less than \(10 \%\) vinegar?

Short answer

After 22 steps, the mixture contains less than 10% vinegar.

Step-by-step solution

Understanding the Problem

We begin with a 1-liter vinegar solution and at each step, we remove 0.1 L of the mixture, replace it with 0.1 L of water, mix, and repeat the process. Our goal is to determine how the concentration of vinegar changes over steps and find when it falls below 10%.

Formulating the Vinegar Concentration After One Step

After removing 0.1 L from 1 L of the original solution, you have 0.9 L of the original mixture. Therefore, the concentration reduction factor is out by 0.1 L each time, or \(0.9\) of the previous concentration.

General Formula for Vinegar Concentration

Given that at each step the concentration reduces by a factor of \(0.9\), the concentration after \(n\) steps is given by \(C_n = C_0 \times 0.9^n\), where \(C_0\) is the initial concentration. Because the process begins with pure vinegar, \(C_0 = 1\). Thus, the concentration formula is \(C_n = 0.9^n\).

Calculate the Number of Steps for Concentration Below 10%

We need to find \(n\) such that \(0.9^n < 0.1\). Taking the natural logarithm on both sides, we have \(n \cdot \ln(0.9) < \ln(0.1)\). Solving for \(n\), we get \(n > \frac{\ln(0.1)}{\ln(0.9)}\).

Numerical Calculation

Using approximate values, \(\ln(0.1) \approx -2.302\) and \(\ln(0.9) \approx -0.105\). Thus, \(n > \frac{-2.302}{-0.105} \approx 21.92\). Since \(n\) must be a whole number, round up to \(n = 22\).

Key concepts

Vinegar Dilution

When dealing with vinegar dilution, we are considering the process of systematically reducing the concentration of vinegar in a solution. Imagine you have 1 liter of pure vinegar. In each step, you remove a portion of it and replace that with water. This step shifts the concentration of vinegar from full strength to something less with each repetition.
To be precise, if you remove 0.1 liter of the mixture (including vinegar and added water) and add back the same amount of water, the new solution contains slightly less vinegar per unit volume. This forms the basis for understanding how dilution impacts overall concentration. With each cycle, the dilution of vinegar becomes more pronounced as more water enters the mixture, altering the composition very gradually, but with cumulative effect.

Concentration Formula

The concentration formula is a mathematical expression that helps quantify how much vinegar remains in the solution after several dilution steps. At the start, vinegar concentration is 1 (or 100% if you like percentages) because we begin with pure vinegar.
After the first dilution, 0.9 liters of the solution remains as vinegar, leading to a concentration reduction factor of 0.9. Hence, the formula for concentration after each subsequent step can be generalized as \( C_n = 0.9^n \), where \( C_n \) denotes the concentration of vinegar after \( n \) dilution steps.
This formula becomes a crucial tool in predicting the vinegar’s concentration without manually performing each dilution step, giving clear insight into the process as you increase the number of iterations.

Step-by-Step Calculation

Conducting a step-by-step calculation is key to tracking the concentration changes and finding specific results, like when vinegar is below a particular concentration threshold. Let's consider this method:

• Start with the initial concentration, \( C_0 \), which is 1 (since the solution is pure vinegar).
• For each dilution step, calculate the new concentration using the established formula \( C_n = 0.9^n \).
• Continue this process incrementally, observing how the vinegar concentration diminishes progressively with the introduction of water.

Following this detailed calculation across multiple steps illuminates the gradual decrease in vinegar concentration, enabling us to answer questions like how many steps are required to achieve less than 10% concentration.

Logarithms in Inequalities

Understanding how to use logarithms in inequalities is essential for solving problems like determining when the vinegar concentration drops below a certain level. In this exercise, our aim is to find the number of dilutions necessary for the concentration to fall below 10%.
To solve this, we use the inequality \( 0.9^n < 0.1 \). Applying logarithms simplifies the algebraic manipulation required, as it transforms multiplicative sequences into additive ones. Thus, we have:
\( n \cdot \,\ln(0.9) < \ln(0.1) \)
From here, calculate \( n \) by comparing common logarithm values: \( \ln(0.1) \approx -2.302 \) and \( \ln(0.9) \approx -0.105 \). Solving gives \( n > \frac{-2.302}{-0.105} \approx 21.92 \). Thus, \( n \) must be at least 22 steps, taking into account whole numbers. This logical use of logarithms in inequalities helps solve for how many steps are required for a specific outcome.