Let's go through the problem.

Given:

The population function is $P(t) = 5.4^t$, and we are asked which of the following statements are always true:

(i) $P'(t) = \lim_{h \to 0} \frac{5.4^h(5.4^t - 1)}{h}$

This expression is trying to represent the derivative of $P(t)$. We'll need to calculate the derivative of $P(t) = 5.4^t$ to see if this limit holds.

(ii) $P'(1)$ is less than the average rate of change of $P$ between 0 and 1.

We need to compute $P'(1)$ and compare it to the average rate of change of $P$ between $t = 0$ and $t = 1$.

(iii) The average rate of change between $0$ and $1$ is 5.4.

We'll compute the average rate of change between $t = 0$ and $t = 1$ and see if it equals 5.4.

Step-by-step process:

1. Find the derivative $P'(t)$: Using the chain rule, the derivative of $P(t) = 5.4^t$ is: $P'(t) = 5.4^t \ln(5.4)$ So, we need to compare this to the limit in (i).

2. Compare the given limit in (i): The limit expression in (i) seems to be an approximation of the derivative using first principles. We'll check if it simplifies to $5.4^t \ln(5.4)$, the correct derivative.

3. Calculate $P'(1)$: Substituting $t = 1$ into the derivative: $P'(1) = 5.4 \ln(5.4)$

4. Average rate of change between $0$ and $1$: The average rate of change between $t = 0$ and $t = 1$ is: $\frac{P(1) - P(0)}{1 - 0} = \frac{5.4^1 - 5.4^0}{1} = 5.4 - 1 = 4.4$

5. Compare $P'(1)$ with the average rate of change: We need to check whether $P'(1)$ is less than the average rate of change, which is 4.4.

Now, let's compute these values to finalize the analysis.Here's what we find:

1. For statement (i): The derivative $P'(t) = 5.4^t \ln(5.4)$ holds true, and the given limit in (i) matches this definition, so this statement is true.

2. For statement (ii): The derivative at $t = 1$, $P'(1) = 9.11$, is greater than the average rate of change (which is $4.4$). Therefore, statement (ii) is false.

3. For statement (iii): The average rate of change between $0$ and $1$ is 4.4, not 5.4, so statement (iii) is false.

Thus, only statement (i) is true. Therefore, the correct answer is:

$\boxed{A}$

Would you like more details on any part of this solution or have further questions?

Here are five related questions to extend your understanding:

1. How would the derivative of a more complex exponential function, such as $P(t) = 7.3^t$, be calculated?
2. Why does the natural logarithm $\ln(5.4)$ appear in the derivative of an exponential function with base 5.4?
3. Can the average rate of change ever be equal to the derivative at a specific point?
4. What does the difference between the average rate of change and the instantaneous rate of change (derivative) tell us about a function's behavior?
5. How would the derivative calculation change if the base of the exponent were Euler's number $e$?

Tip: The derivative of an exponential function $a^t$ is given by $a^t \ln(a)$, which arises from the chain rule and the properties of logarithms.