To solve this, let's work through the two parts provided:

Given:

The exponential function is: $f(x) = 574 \cdot (1.026)^x$ Where $x$ is the number of years after 1974.

Part (a):

Substitute $x = 0$:

When $x = 0$, the equation becomes: $f(0) = 574 \cdot (1.026)^0$

Since $(1.026)^0 = 1$, we have: $f(0) = 574$

So, the population in 1974 is 574 million.

Part (b):

Substitute $x = 27$ (for the year 2001):

Now substitute $x = 27$ into the formula: $f(27) = 574 \cdot (1.026)^{27}$

1. Calculate $(1.026)^{27}$: Using a calculator: $(1.026)^{27} \approx 2.093$

2. Multiply by 574: $f(27) = 574 \cdot 2.093 \approx 1201$

Thus, the population in 2001 is approximately 1,201 million.

Would you like further details or have any questions? Here are five related questions for exploration:

1. How does exponential growth differ from linear growth in population modeling?
2. Can we use this model to predict the population in 2050?
3. What is the percentage growth rate per year in this model, and how does it relate to the multiplier $1.026$?
4. What are the limitations of using exponential functions for population predictions over a long period?
5. How can this population model be adjusted for real-world factors like economic changes or disasters?

Tip: Exponential functions are ideal for modeling growth when a quantity grows at a constant percentage rate per unit time.