This question involves rewriting the population function $P(t) = 20(2.50)^{\frac{t}{5}}$ in an equivalent form to reveal the annual growth rate. Let's solve this step by step:

Step 1: Rewrite the function

The given function is:

$P(t) = 20(2.50)^{\frac{t}{5}}.$

Using the property of exponents, $a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m$, we can rewrite:

$(2.50)^{\frac{t}{5}} = \left((2.50)^{\frac{1}{5}}\right)^t.$

Step 2: Simplify the base

Calculate $(2.50)^{\frac{1}{5}}$, which is the fifth root of 2.50. Approximating this gives:

$(2.50)^{\frac{1}{5}} \approx 1.20.$

Step 3: Write the equivalent form

Substituting $(2.50)^{\frac{1}{5}}$ back, the function becomes:

$P(t) = 20(1.20)^t.$

Final Answer:

The equivalent form is:

$P(t) = 20(1.20)^t.$

This reveals the annual growth rate of the fish population is 1.20, which corresponds to a 20% annual growth rate.

Would you like further explanation or details about the steps?

Follow-up Questions:

1. How does the property of exponents help in solving this problem?
2. Why is the annual growth rate represented by the base $1.20$?
3. How can the fifth root be computed more accurately if necessary?
4. What is the significance of rewriting $P(t)$ in this form for real-world applications?
5. How does this model change if the growth rate is reduced or increased?

Tip:

When rewriting exponential functions, focus on isolating the base to reveal growth or decay rates easily!