Let's break down the problem you uploaded.

1) Exponential Regression

We are asked to find a function of the form:

$P(t) = a \cdot b^t$

Where:

• $P(t)$ is the population in billions,
• $t$ is the number of years since 1900 (so, for example, $t = 50$ for 1950),
• $a$ and $b$ are constants to be found using exponential regression.

Given data points (Year, Population in Billions):

• (0, 1.7) for 1900
• (50, 2.5) for 1950
• (99, 6.0) for 1999
• (111, 7.0) for 2011

To find values for $a$ and $b$, we typically use a calculator or software to perform exponential regression on this dataset.

2) Future Population Estimates

Once we have $a$ and $b$, we can use the model to predict the population for the years 2050 and 2100. For these years:

• $t = 150$ for 2050 (since it's 150 years after 1900),
• $t = 200$ for 2100.

3) Growth Rate in 2050 and 2100

We can compute the growth rate using the derivative of the exponential model. The rate of growth at any time $t$ can be found as:

$\text{Growth rate} = P'(t) = \ln(b) \cdot a \cdot b^t$

Given the values for $a$ and $b$, we calculate the rates for the years 2050 and 2100.

Would you like me to help you perform the regression to find $a$ and $b$, or do you have access to a calculator for that?

Here are a few follow-up questions that expand on this:

1. Can you explain how exponential models differ from linear models in terms of population growth?
2. What might cause actual population growth to deviate from this model in real life?
3. How does changing the base $b$ in the equation $P(t) = a \cdot b^t$ affect the long-term population estimates?
4. Why do we use natural logarithms ($\ln$) when calculating growth rates from exponential models?
5. Could you derive the exponential growth rate formula if given a different type of growth function?

Tip: When performing exponential regression on a calculator, it's important to enter the data points correctly to avoid errors in $a$ and $b$.