To solve this problem, we will use the rule of 70 for approximate doubling time. This rule states:

$\text{Doubling Time} = \frac{70}{\text{Growth Rate (\%)}}.$

Step 1: Calculate the doubling time

Given an annual growth rate of $0.9\%$:

$\text{Doubling Time} = \frac{70}{0.9} \approx 77.8 \, \text{years}.$

Step 2: Calculate population at various ages

We start with the population in 2021: $7.8 \, \text{billion}$. The student was born in 2001, so their age in any given year is:

$\text{Age in a year} = \text{Year} - 2001.$

Part 1: Population at age 50

When the student is $50$ years old, the year is:

$2001 + 50 = 2051.$

From 2021 to 2051 is $30$ years. The population change can be estimated using the formula for exponential growth:

$P(t) = P_0 \cdot 2^{t / \text{Doubling Time}},$ where:

• $P_0 = 7.8 \, \text{billion}$,
• $t = 2051 - 2021 = 30 \, \text{years}$,
• $\text{Doubling Time} = 77.8 \, \text{years}$.

$P(30) = 7.8 \cdot 2^{30 / 77.8}.$

$P(30) = 7.8 \cdot 2^{0.3855} \approx 7.8 \cdot 1.307 \approx 10.2 \, \text{billion}.$

So, at age $50$, the population will be approximately 10.2 billion.

Part 2: Population at age 83

When the student is $83$ years old, the year is:

$2001 + 83 = 2084.$

From 2021 to 2084 is $63$ years. Using the same formula:

$P(63) = 7.8 \cdot 2^{63 / 77.8}.$

$P(63) = 7.8 \cdot 2^{0.8098} \approx 7.8 \cdot 1.749 \approx 13.6 \, \text{billion}.$

So, at age $83$, the population will be approximately 13.6 billion.

Part 3: Population at age 100

When the student is $100$ years old, the year is:

$2001 + 100 = 2101.$

From 2021 to 2101 is $80$ years. Using the same formula:

$P(80) = 7.8 \cdot 2^{80 / 77.8}.$

$P(80) = 7.8 \cdot 2^{1.028} \approx 7.8 \cdot 2.054 \approx 16.0 \, \text{billion}.$

So, at age $100$, the population will be approximately 16.0 billion.

---

Final Results

1. When the student is 50 years old (2051), the population will be about 10.2 billion.
2. When the student is 83 years old (2084), the population will be about 13.6 billion.
3. When the student is 100 years old (2101), the population will be about 16.0 billion.

Would you like a detailed explanation of any step?

---

Here are 5 related questions for further exploration:

1. How does the rule of 70 compare to exact exponential growth calculations?
2. What happens if the annual growth rate decreases in future decades?
3. How would this calculation change with compounding growth rates (e.g., variable growth)?
4. How can we model population changes in regions with declining populations?
5. What is the significance of the doubling time in real-world scenarios like resource management?

Tip: Always verify population models using multiple methods, especially for long-term predictions where assumptions may change.