Problem 46 - College Algebra

Question

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(70 \% ?\)

Step-by-Step Solution

Answer

Therefore, the age at which the percentage of Americans with coronary heart disease is 70% is \(x = -\frac{ln(\frac{20}{271})}{0.122}\) years.

Step 1: Understand the task

This problem involves finding the value of 'x', which represents the age, when the percentage, \(P(x)\), of Americans with coronary heart disease is 70. Given the logistic growth function \(P(x) = \frac{90}{1+271e^{-0.122x}}\) we're tasked with finding 'x' when \(P(x) = 70\).

Step 2: Set up the equation

Replace \(P(x)\) in the logistic growth function with 70, which gives \(70 = \frac{90}{1+271e^{-0.122x}}\)

Step 3: Simplify the equation

Rearrange the equation to isolate the exponential part. Multiply both sides by \(1+271e^{-0.122x}\) and then subtract 70 from both sides. This gives \(271e^{-0.122x} = 90 - 70\)

Step 4: Further simplify the equation

Simplify the right side of the equation to get \(271e^{-0.122x} = 20\). Then, divide both sides by 271 to isolate \(e^{-0.122x}\), resulting in the equation \(e^{-0.122x} = \frac{20}{271}\)

Step 5: Solve the equation

We can now convert this equation to log form to make it easier to solve. Using the property of natural logarithms \(ln(e^a) = a\), we can express this as \(ln(e^{-0.122x}) = ln(\frac{20}{271})\), which simplifies to \(-0.122x = ln(\frac{20}{271})\)

Step 6: Final solution

Solve for x, by first negating both sides and then dividing by 0.122, which gives \(x = -\frac{ln(\frac{20}{271})}{0.122}\)

Key Concepts

Coronary Heart Disease Modeling

Modeling diseases like coronary heart disease using mathematical functions helps in understanding their spread and impact on different age groups. In this case, we use a logistic growth function, which is effective because:

• It limits the growth to a capacity, recognizing that not everyone will be affected.
• It considers the impact of various factors such as age and risk thresholds.

The function given, \[ P(x) = \frac{90}{1 + 271 e^{-0.122x}} \] helps in calculating the percentage of Americans affected by coronary heart disease for different ages \(x\). Such models allow medical researchers and policymakers to project healthcare needs over time.

Percentage Calculation

Understanding percentage calculations in models like this is crucial. Here, we're interested in the percentage of people with coronary heart disease at different ages. The logistic function provides this as:

• \(P(x)\), the percentage, varies with \(x\), the age.
• It helps in making predictions about population health over time.

To solve the problem, we want to find when \(P(x) = 70\). Setting up the equation: \[70 = \frac{90}{1 + 271 e^{-0.122x}}\] allows us to determine the age at which this condition occurs, giving valuable insights into when interventions might be necessary.

Solving Exponential Equations

Exponential equations often appear in growth models, and understanding how to solve them is essential. Here is the breakdown of solving:

• Start with the equation \(70 = \frac{90}{1 + 271 e^{-0.122x}}\).
• Reorganize to isolate the exponential term: \( 271e^{-0.122x} = 20 \).
• Divide by 271 to ensure \( e^{-0.122x} \) stands alone, giving \( e^{-0.122x} = \frac{20}{271} \).

This process involves simplifying and re-arranging, which is crucial to finding the variable hidden within exponentials.

Natural Logarithms in Algebra

Using natural logarithms effectively can simplify solving exponential equations. Here's how they work:

• The natural logarithm \(ln\) is the inverse of the exponential function \(e^x\).
• For a term like \(e^{-0.122x} = \frac{20}{271}\), taking the natural log gives \(ln(e^{-0.122x}) = ln(\frac{20}{271})\).
• This simplifies to \(-0.122x = ln(\frac{20}{271})\).

To solve for \(x\), divide by \(-0.122\) to isolate it, resulting in the final solution: \[x = -\frac{ln(\frac{20}{271})}{0.122}\] Utilizing natural logarithms can make complex problems more manageable and easier to compute.