Problem 96 - College Algebra with Modeling and Visualization

Question

Filters Impurities in water are frequently removed using filters. Suppose that a 1 -inch filter allows \(10 \%\) of the impurities to pass through it. The other \(90 \%\) is trapped in the filter. (a) Find a formula in the form \(f(x)=100 a^{x}\) that calculates the percentage of impurities passing through \(x\) inches of this type of filter. (b) Use \(f(x)\) to estimate the percentage of impurities passing through 2.3 inches of the filter.

Step-by-Step Solution

Answer

The formula is \(f(x) = 100 \cdot (0.10)^x\); for 2.3 inches, \(f(2.3) \approx 0.501\%\) of impurities pass through.

Step 1: Understand the Problem

We need to calculate the percentage of impurities passing through different thicknesses of a filter. Given that for a 1-inch filter, 10% of the impurities pass through and 90% are trapped, we need a general formula for any inch of the filter.

Step 2: Identify the Exponential Decay

Since the rate of impurities passing through decreases exponentially with each inch of filter, identify a base that represents this decrease. The problem states 10% passes per inch, so 0.10 of the impurities pass and 0.90 is trapped per inch.

Step 3: Formulate the Equation

The exponential model for decay can be written as \(f(x) = 100 \cdot a^{x}\), where \(a\) is the fraction of impurities passing through each subsequent inch. Since 10% passes in the first inch, set \(a = 0.10\). Thus, our formula is \(f(x) = 100 \cdot (0.10)^x\).

Step 4: Solve for the Specific Thickness

To find the percentage of impurities passing through 2.3 inches of the filter, substitute \(x = 2.3\) into the equation: \[f(2.3) = 100 \cdot (0.10)^{2.3}.\]

Step 5: Calculate the Exact Value

Using a calculator, evaluate \((0.10)^{2.3} \approx 0.00501\). Then, multiply by 100 to find: \[f(2.3) = 100 \cdot 0.00501 = 0.501\].

Step 6: Interpret the Result

The result means that approximately 0.501% of the impurities pass through a 2.3-inch filter. This demonstrates the effectiveness of increasing filter thickness.

Key Concepts

Exponential Function

Exponential functions are a key component in modeling various real-world scenarios, including the phenomenon of exponential decay. These functions are expressed in the form \(f(x) = a \cdot b^{x}\), where:

• \(a\) represents the initial amount or starting value.
• \(b\) is the base of the exponential function, reflecting the growth or decay rate.

In this specific exercise about water filters, the function's form is \(f(x) = 100 \cdot a^{x}\), designed to reflect the decay of impurities passing through different thicknesses of the filter. With each layer of filtration, a smaller percentage gets through, highlighting a decrease - an exponential decay - due to the consistent multiplicative rate that the filters provide. Here, the base \(a\) is particularly important. Since only 10% of impurities pass through a 1-inch filter, \(a=0.10\), demonstrating how much impurities continue beyond each additional inch. This specific function form is excellent for showcasing steady decline in passing impurities as filter thickness increases.

Algebraic Modeling

Algebraic modeling provides a structured approach to represent real-life problems with mathematical expressions. It helps in visualization and solving various practical issues through equations and graphs.

In the context of this exercise, algebraic modeling comes into play by creating a function to estimate impurities passing through different thicknesses of a filter. The given condition, that 10% of impurities pass for each inch, sets up the stage for modeling this with an exponential decay function \(f(x) = 100 \cdot (0.10)^x\).

Having this function allows one to easily calculate the behavior of impurities for any filter thickness by simply substituting the desired thickness into the function as \(x\). This powerful tool not only helps in predicting outcomes but also in visualizing how changes in one parameter (like filter thickness) impact the results (percentage of passing impurities). By formulating the problem algebraically, engineers, scientists, and students can choose filter designs that maximize efficiency based on mathematical predictions.

Problem-Solving Steps

Solving a problem involving exponential decay requires a series of logical steps. Let's break them down for clarity.

• Step 1: Understand the Problem
  This involves identifying what needs solving. For our water filter example, it's about finding how much impurity passes as filter thickness increases.
• Step 2: Recognize the Exponential Aspect
  The behavior of impurity passage is exponential decay since with each additional inch, the impurities passing decrease at a rate of the previous inch's 10%.
• Step 3: Formulate the Equation
  Establish the function using the initial conditions given, \(f(x) = 100 \cdot (0.10)^{x}\), where 10% passage is known for one inch of filter.
• Step 4: Apply the Formula
  Use this encapsulated equation for any thickness - substituting directly into the function. In our case, inputting \(x =2.3\) yields the specific solution.
• Step 5: Calculate
  Use a calculator to evaluate the specific values, such as \(f(2.3) = 0.501\), to interpret the practical implication.
• Step 6: Interpret the Results
  This means contextualizing the outcome to see what the numbers say - how effective the filter is at such thickness in letting impurities through, or blocking them.

By following these steps, anyone can tackle this exercise with confidence, understanding not only how to calculate specific instances but also how to apply these principles to similar scenarios.