Given the exponential function for the population size of a fish species in a lake:
[tex]\[ P(t) = 340 \cdot (1.22)^t \][/tex]
Let's answer each part of the question step by step.
1. Find the initial population size.
The initial population size corresponds to the population at time [tex]\( t = 0 \)[/tex]. To find this, we substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[ P(0) = 340 \cdot (1.22)^0 \][/tex]
Any number raised to the power of 0 is 1, so:
[tex]\[ P(0) = 340 \cdot 1 = 340 \][/tex]
Therefore, the initial population size is [tex]\( \boxed{340} \)[/tex].
2. Does the function represent growth or decay?
To determine whether the function represents growth or decay, we need to look at the base of the exponential term, which is 1.22. If the base is greater than 1, the function represents growth. If the base is between 0 and 1, it represents decay.
In this case, the base 1.22 is greater than 1. Therefore, the function represents [tex]\( \boxed{\text{growth}} \)[/tex].
3. By what percent does the population size change each year?
The percentage change each year is determined by the factor by which the population grows. Since the base of the exponential term is 1.22, this means that the population grows by a factor of 1.22 each year.
To find the percentage change, we subtract 1 from the base and then convert it to a percentage:
[tex]\[ (1.22 - 1) \times 100 \% = 0.22 \times 100 \% = 22 \% \][/tex]
Hence, the population size changes by [tex]\( \boxed{22} \% \)[/tex] each year.
