Let's analyze the given problem step by step:
### Initial Population Size
The population size [tex]\( P(t) \)[/tex] is described by the function:
[tex]\[ P(t) = 550 \cdot (0.78)^t \][/tex]
To find the initial population size, we evaluate the function at [tex]\( t = 0 \)[/tex]:
[tex]\[ P(0) = 550 \cdot (0.78)^0 \][/tex]
Since any number raised to the power of zero is 1, we get:
[tex]\[ P(0) = 550 \cdot 1 = 550 \][/tex]
So, the initial population size is:
[tex]\[ \boxed{550.0} \][/tex]
### Growth or Decay
To determine if the function represents growth or decay, we examine the base of the exponential function, which is [tex]\( 0.78 \)[/tex].
- If the base is less than 1, it represents decay.
- If the base is greater than 1, it represents growth.
Here, the base [tex]\( 0.78 \)[/tex] is less than 1, which means the function represents decay. Therefore, we select:
[tex]\[ \boxed{\text{decay}} \][/tex]
### Percent Change Each Year
To find the percent change in the population size each year, we first calculate the population after 1 year:
[tex]\[
P(1) = 550 \cdot (0.78)^1 = 550 \cdot 0.78
\][/tex]
Next, we compute the percent change each year. The percent change is determined by the difference between the initial population and the population after 1 year relative to the initial population:
[tex]\[
\text{Percent change per year} = \left(\frac{\text{Initial population} - \text{Population after 1 year}}{\text{Initial population}}\right) \times 100
\][/tex]
Using the values we have:
[tex]\[
\text{Initial population} = 550
\][/tex]
[tex]\[
\text{Population after 1 year} = 550 \cdot 0.78 = 429
\][/tex]
The percent change is:
[tex]\[
\text{Percent change per year} = \left(\frac{550 - 429}{550}\right) \times 100 = \left(\frac{121}{550}\right) \times 100 \approx 22\%
\][/tex]
So, the population size changes by:
[tex]\[ \boxed{22.0\%} \][/tex]
This completes our step-by-step solution. The initial population size is 550, the function represents decay, and the population size decreases by 22.0% each year.
