Question

a logistic growth model for world population, ( f(x) ), in billions, ( x ) years after 1971 is ( f(x)=\frac{12.57}{1 + 4.11e^{-0.026x}} ). according to this model, when will the world population be 8 billion? according to this model, the world population will be 8 billion in ( square ). (round to the nearest whole number as needed.)

Explanation:

Step1: Substitute \( f(x) = 8 \) into the function

We have the equation \( 8=\frac{12.57}{1 + 4.11e^{-0.026x}}\).
First, cross - multiply: \(8(1 + 4.11e^{-0.026x})=12.57\).
Then, distribute: \(8+32.88e^{-0.026x}=12.57\).
Subtract 8 from both sides: \(32.88e^{-0.026x}=12.57 - 8=4.57\).
Divide both sides by 32.88: \(e^{-0.026x}=\frac{4.57}{32.88}\approx0.139\).

Step2: Take the natural logarithm of both sides

Since \(y = e^{x}\) and \(x=\ln(y)\) are inverse functions, if \(e^{-0.026x}=0.139\), then \(-0.026x=\ln(0.139)\).
We know that \(\ln(0.139)\approx - 1.977\).
So, \(x=\frac{-1.977}{-0.026}\approx76\).

Answer:

1971 + 76 = 2047