Let’s break this problem into its components and solve step by step.

Given:

The model is $P = ab^t$, where:

• $P$ is the population in billions.
• $t$ is the number of years after 2004.
• The log-transformed version is $\log_{10}P = \log_{10}a + t\log_{10}b$, showing a linear relationship.

From the question:

• Gradient of the line ($\log_{10}b$) = 0.0054
• Intercept on the $\log_{10}P$-axis ($\log_{10}a$) = 0.81

(a) Estimating $a$ and $b$:

1. Determine $a$:

The intercept $\log_{10}a = 0.81$: $a = 10^{0.81}$ Calculating: $a \approx 6.457$

2. Determine $b$:

The gradient $\log_{10}b = 0.0054$: $b = 10^{0.0054}$ Calculating: $b \approx 1.012$

Thus:

• $a \approx 6.457$
• $b \approx 1.012$

(b) Interpretation of the constants:

(i) $a$:

The constant $a$ represents the world population (in billions) in 2004, as $t = 0$ in that year. Hence, $a \approx 6.457$ billion.

(ii) $b$:

The constant $b$ represents the annual growth factor. A value of $b \approx 1.012$ indicates a $1.2\%$ annual increase in population.

(c) Estimate the population in 2030:

1. Time difference:

2030 corresponds to $t = 2030 - 2004 = 26$ years.

2. Model substitution:

$P = ab^t = 6.457 \times (1.012)^{26}$ Calculating: $P \approx 6.457 \times 1.34885 \approx 8.706 \, \text{billion}$

Thus, the estimated population in 2030 is approximately 8.706 billion.

(d) Comment on reliability:

The reliability of this estimate depends on:

1. Assumption of a consistent growth rate ($b$) over time, which may not hold due to socioeconomic and environmental factors.
2. The model is based on data from only 2004 to 2007, making long-term predictions less reliable.
3. External factors like global crises or policy changes are not accounted for, potentially affecting population growth.

Thus, while the model provides a mathematical estimate, its reliability for 2030 is limited.

Do you have further questions about this solution? Here are five related questions for further exploration:

1. How does changing the gradient ($\log_{10}b$) affect the population prediction?
2. What would the model predict for a shorter or longer time period, such as 2015 or 2100?
3. How could incorporating additional data points improve the model's reliability?
4. What alternative growth models could be used to predict population changes?
5. How does the percentage growth rate relate to $b$ and $t$?

Tip: Always consider external factors (e.g., technology, policy changes) when interpreting long-term predictions from mathematical models.