Problem 45 - Algebra and Trigonometry

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 \(50 \% ?\)

Step-by-Step Solution

Answer

The age at which 50% of Americans are expected to have some coronary heart disease is approximately \(x = \frac{\ln \left( \frac{\frac{90}{50}-1}{271} \right)}{-0.122}\) years.

Step 1: Set up the Equation

Set \(P(x)\) equal to 50 (because we are looking for the age when the percentage is 50%) and solve for \(x\). \[ 50 = \frac{90}{1+271e^{-0.122x}} \]

Step 2: Isolate the Exponential Part

To isolate the exponential part, multiply both sides of the equation by \(1+271e^{-0.122x}\) and then subtract 50 from both sides. \[ 50(1+271e^{-0.122x}) = 90 \]\[ 271e^{-0.122x} = \frac{90}{50} - 1 \]

Step 3: Solve for the Exponent

To solve for the exponent, we will divide both sides of the equation by 271 and then take the natural logarithm of both sides to get rid of the exponential. \[ e^{-0.122x} = \frac{\frac{90}{50}-1}{271} \]\[ -0.122x = \ln \left( \frac{\frac{90}{50}-1}{271} \right) \]

Step 4: Solve for 'x'

To find the value of \(x\), the age at which 50% of Americans have coronary heart disease, divide both sides of the equation by \(-0.122\). \[ x = \frac{\ln \left( \frac{\frac{90}{50}-1}{271} \right)}{-0.122} \]