Problem 46

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

Verified

Answer

The age at which the percentage of Americans with some coronary heart disease is $70 \%$ cannot be determined from the given function, as it results in taking the logarithm of a negative number, which is undefined in real numbers. Our logistic function may not accurately represent this percentage beyond its certain range.

1Step 1: Set P(x) to 70

Insert $70$ for $P(x)$ in our function: \[ 70 = \frac{90}{1+271e^{-0.122x}} \]

2Step 2: Rearrange the equation to isolate the exponential

Rearrange the equation to isolate $271 e^{-0.122x}$:\[ 271 e^{-0.122x} = \frac{90}{70} - 1 \]

3Step 3: Simplify

Calculate $\frac{90}{70} - 1$ and simplify:\[ 271 e^{-0.122x} = 0.285714286 - 1 = -0.714285714\]

4Step 4: Solve for e^{-0.122x}

Divide both sides by 271 to get $e^{-0.122x}$:\[ e^{-0.122x} = \frac{-0.714285714}{271} \]

5Step 5: Take the natural log of both sides

Taking the natural log, we get:\[ -0.122x = \ln |\frac{-0.714285714}{271}| \]

6Step 6: Solve for x

Finally solve the equation for $x$. Divide both sides by -0.122 to get:\[ x = \frac{\ln |\frac{-0.714285714}{271}|}{-0.122} \]