Let's now revisit the earth population problem in question 13, and update the numbers given there. Let t  be the number of years since 1975. Let  y(t) be the function computed in the previous problem (question 14). Some scientists believe that 16 billion is a reasonable estimate of the carrying capacity of earth. So let's assume that m=0 and M=16
To get round numbers let's assume that the population in 1975 (t=0) was 4 billion, and in 2024  (t=49)  it was 8 billion. From this information you can compute 
k=
   (Enter your answer with at least 6 correct digits beyond the decimal point.) Don't reenter the rounded value of  k
 into your calculator. Instead store the computed value and use it for the next question.)

Using the computed value of k you can estimate that 30 years later, in the year 2054 (t=79), the population of earth will be ? billion. (Enter your answer with at least three correct digits beyond the decimal point.)

We are given the following:

- The carrying capacity \( M = 16 \) billion.
- Population at \( t = 0 \) (in 1975) is \( P(0) = 4 \) billion.
- Population at \( t = 49 \) (in 2024) is \( P(49) = 8 \) billion.
- We want to model the population growth using a logistic growth function.

### Logistic Growth Model:
The population \( P(t) \) as a function of time \( t \) can be modeled by the equation:

\[
P(t) = \frac{M}{1 + \left( \frac{M}{P_0} - 1 \right) e^{-kt}}
\]
where:
- \( M \) is the carrying capacity (in billions),
- \( P_0 \) is the initial population (at \( t = 0 \)),
- \( k \) is the growth rate constant,
- \( t \) is the number of years since 1975.

We need to find the value of \( k \) first.

### Step 1: Set up the equation for \( t = 49 \) (year 2024):
At \( t = 49 \), the population is \( P(49) = 8 \) billion. Substituting into the logistic equation:

\[
8 = \frac{16}{1 + \left( \frac{16}{4} - 1 \right) e^{-49k}}
\]

Simplifying this:

\[
8 = \frac{16}{1 + (4 - 1) e^{-49k}}
\]
\[
8 = \frac{16}{1 + 3 e^{-49k}}
\]
Multiply both sides by \( 1 + 3e^{-49k} \):

\[
8(1 + 3e^{-49k}) = 16
\]
\[
8 + 24e^{-49k} = 16
\]
\[
24e^{-49k} = 8
\]
\[
e^{-49k} = \frac{8}{24} = \frac{1}{3}
\]

Taking the natural logarithm on both sides:

\[
-49k = \ln\left(\frac{1}{3}\right)
\]
\[
k = \frac{-\ln(1/3)}{49}
\]

Now, let’s compute \( k \).The value of \( k \) is approximately \( 0.022420659 \).

### Step 2: Estimate the population in 2054 (\( t = 79 \)):

Now that we have \( k \), we can estimate the population in 2054, where \( t = 79 \). Using the logistic growth formula again:

\[
P(79) = \frac{16}{1 + \left( \frac{16}{4} - 1 \right) e^{-79k}}
\]

Substitute the values:

\[
P(79) = \frac{16}{1 + 3e^{-79k}}
\]

Let's compute \( P(79) \).The estimated population of Earth in the year 2054, using the logistic growth model, is approximately \( 10.593 \) billion.

Do you have any further questions, or would you like additional details?

Here are 5 follow-up questions you might consider:

1. How does the carrying capacity \( M \) affect the long-term behavior of population growth?
2. How would changing the initial population affect the value of \( k \)?
3. What happens to the population growth rate as \( t \) approaches infinity in this model?
4. How sensitive is the population estimate in 2054 to small changes in \( k \)?
5. What other factors might influence real-world population growth beyond this model?

**Tip:** Always keep enough significant digits when calculating exponential functions to maintain accuracy over long timescales.

Let's now revisit the earth population problem in question 13, and update the numbers given there. Let t be the number of years since 1975. Let y(t) be the function computed in the previous problem (question 14). Some scientists believe that 16 billion is a reasonable estimate of the carrying capacity of earth. So let's assume that m=0 and M=16. To get round numbers let's assume that the population in 1975 (t=0) was 4 billion, and in 2024 (t=49) it was 8 billion. From this information you can compute k= (Enter your answer with at least 6 correct digits beyond the decimal point.) Don't reenter the rounded value of k into your calculator. Instead store the computed value and use it for the next question. Using the computed value of k you can estimate that 30 years later, in the year 2054 (t=79), the population of earth will be ? billion. (Enter your answer with at least three correct digits beyond the decimal point.)

Explore the logistic growth model to estimate Earth's population in 2054. With a carrying capacity of 16 billion, learn how to compute the growth rate constant (k) and apply it to predict future population levels. This problem covers population modeling, exponential functions, and real-world applications.

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