Question

a logistic growth model for world population, f(x), in billions, x years after 1977 is $f(x) = \frac{12.57}{1 + 4.11e^{-0.026x}}$. according to this model, when will the world population be 11 billion?
according to this model, the world population will be 11 billion in \square.
(round to the nearest whole number as needed.)

Explanation:

Step1: Set up the equation

We know that \( f(x) = 11 \), and the function is \( f(x)=\frac{12.57}{1 + 4.11e^{-0.026x}} \). So we set up the equation:

$$ 11=\frac{12.57}{1 + 4.11e^{-0.026x}} $$

Step2: Solve for the denominator

Multiply both sides by \( 1 + 4.11e^{-0.026x} \):

$$ 11(1 + 4.11e^{-0.026x})=12.57 $$

Then divide both sides by 11:

$$ 1 + 4.11e^{-0.026x}=\frac{12.57}{11}\approx1.1427 $$

Step3: Isolate the exponential term

Subtract 1 from both sides:

$$ 4.11e^{-0.026x}=1.1427 - 1 = 0.1427 $$

Then divide both sides by 4.11:

$$ e^{-0.026x}=\frac{0.1427}{4.11}\approx0.0347 $$

Step4: Take the natural logarithm

Take the natural log of both sides:

$$ \ln(e^{-0.026x})=\ln(0.0347) $$

Since \( \ln(e^a)=a \), we get:

$$ -0.026x=\ln(0.0347)\approx - 3.367 $$

Step5: Solve for x

Divide both sides by - 0.026:

$$ x=\frac{-3.367}{-0.026}\approx129.5 $$

Rounding to the nearest whole number, \( x\approx130 \). Now we find the year by adding 130 to 1977: \( 1977 + 130 = 2107 \).

Answer:

2107