Starting from the estimated 7.8
 
billion world population in​ 2021, assume that the population maintains an annual growth rate of 0.9
​%. 
Suppose a student was born in 2001
. 
What will the world population be when the student is 45
 
years​ old? 78
 
years​ old? 98
 
years​ old? Use the approximate doubling time formula.


To solve this problem, we can use the **rule of 70** for estimating population doubling time and then calculate the population at different ages of the student.

### Step-by-Step Explanation:

#### 1. **Doubling Time Formula**
The **rule of 70** gives us the approximate doubling time of a population based on its growth rate. The formula is:

\[
\text{Doubling Time} = \frac{70}{\text{Annual Growth Rate (\%)}}
\]

Here, the annual growth rate is 0.9%, so:

\[
\text{Doubling Time} = \frac{70}{0.9} \approx 77.78 \text{ years}
\]

This means the world population will approximately double every 77.78 years.

#### 2. **Population Growth Formula**
For any given number of years, \( t \), after the initial population \( P_0 \), we can use the population growth formula:

\[
P(t) = P_0 \times 2^{\frac{t}{\text{Doubling Time}}}
\]

Where:
- \( P(t) \) is the population after time \( t \) years,
- \( P_0 = 7.8 \text{ billion} \) (the population in 2021),
- \( t \) is the number of years from 2021 to the target year,
- Doubling Time \( \approx 77.78 \text{ years} \).

#### 3. **Age 45 Calculation**
A student born in 2001 will be 45 years old in \( 2001 + 45 = 2046 \).

So, the number of years from 2021 to 2046 is:

\[
t = 2046 - 2021 = 25 \text{ years}
\]

Now, we can calculate the population in 2046:

\[
P(25) = 7.8 \times 2^{\frac{25}{77.78}} \approx 7.8 \times 2^{0.3215} \approx 7.8 \times 1.253 \approx 9.77 \text{ billion}
\]

#### 4. **Age 78 Calculation**
A student born in 2001 will be 78 years old in \( 2001 + 78 = 2079 \).

So, the number of years from 2021 to 2079 is:

\[
t = 2079 - 2021 = 58 \text{ years}
\]

Now, we calculate the population in 2079:

\[
P(58) = 7.8 \times 2^{\frac{58}{77.78}} \approx 7.8 \times 2^{0.7456} \approx 7.8 \times 1.678 \approx 13.09 \text{ billion}
\]

#### 5. **Age 98 Calculation**
A student born in 2001 will be 98 years old in \( 2001 + 98 = 2099 \).

So, the number of years from 2021 to 2099 is:

\[
t = 2099 - 2021 = 78 \text{ years}
\]

Now, we calculate the population in 2099:

\[
P(78) = 7.8 \times 2^{\frac{78}{77.78}} \approx 7.8 \times 2^{1.0028} \approx 7.8 \times 2.01 \approx 15.68 \text{ billion}
\]

### Final Answers:
- When the student is 45 years old (in 2046), the population will be approximately **9.77 billion**.
- When the student is 78 years old (in 2079), the population will be approximately **13.09 billion**.
- When the student is 98 years old (in 2099), the population will be approximately **15.68 billion**.

---

Would you like more details on any step?

Here are 5 related questions:
1. How accurate is the doubling time formula for real-world population projections?
2. How does varying the growth rate impact the doubling time?
3. What factors could cause the population growth rate to change over time?
4. How would you calculate population decline using similar formulas?
5. How does the logistic growth model differ from exponential growth?

**Tip:** The rule of 70 is a quick estimation method but doesn't account for fluctuations in growth rates over long periods!

Starting from the estimated 7.8 billion world population in 2021, assume that the population maintains an annual growth rate of 0.9%. Suppose a student was born in 2001. What will the world population be when the student is 45 years old? 78 years old? 98 years old? Use the approximate doubling time formula.

Estimate the future world population using the rule of 70 for exponential growth. Starting from 7.8 billion in 2021, this problem calculates the population when a student born in 2001 turns 45, 78, and 98 years old, considering a 0.9% annual growth rate. Learn the application of the doubling time formula in population projections.

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