A study is done on the population of a certain fish species in a lake. Suppose that the population size $$ P(t) $$ after $$ t $$ years is given by the following exponential function. $$ P(t)=530(1.36)^{t} $$ Find the initial population size. Does the function represent growth or decay? Browth $$ \quad $$ o decay By percent does the population size change each year? $$ \square \% $$

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1. **Initial Population Size**

The population function is given by
   $$
   P(t)=530(1.36)^t.
   $$
   To find the initial population size, we evaluate the function at $$ t=0 $$:
   $$
   P(0)=530(1.36)^0=530\cdot 1=530.
   $$
   Thus, the initial population size is $$ 530 $$ fish.

2. **Growth or Decay**

The base of the exponential function is $$ 1.36 $$. Since $$ 1.36>1 $$, the function represents exponential growth.

3. **Annual Percent Change**

The growth factor each year is $$ 1.36 $$, meaning the population multiplies by $$ 1.36 $$ each year. The percent change is given by
   $$
   (1.36-1)\times 100\% = 0.36 \times 100\% = 36\%.
   $$

Therefore, the population increases by $$ 36\% $$ each year.
- **Initial Population Size:** 530 fish - **Growth or Decay:** Growth - **Annual Percent Change:** 36%

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