To determine the correct model that approximates the number of wasps in the colony for any given week, we need to understand how exponential growth works, given the initial population and the growth rate.
Step-by-Step Solution:
1. Initial Population:
The colony starts with 4 wasps. This is our initial value, denoted by [tex]\( P_0 \)[/tex].
2. Growth Rate:
The number of wasps grows by 32% each week. To express this mathematically, we convert the percentage to a decimal: 32% = 0.32. This means each week, the population grows by a factor of [tex]\( 1 + 0.32 = 1.32 \)[/tex].
3. Exponential Growth Model:
With exponential growth, the population after [tex]\( x \)[/tex] weeks can be expressed as:
[tex]\[
P(x) = P_0 \cdot (1 + r)^x
\][/tex]
where:
- [tex]\( P(x) \)[/tex] is the population after [tex]\( x \)[/tex] weeks,
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate per week,
- [tex]\( x \)[/tex] is the number of weeks.
4. Population after [tex]\( x \)[/tex] weeks:
Substituting the initial population [tex]\( P_0 = 4 \)[/tex] and the growth rate [tex]\( r = 0.32 \)[/tex]:
[tex]\[
P(x) = 4 \cdot (1.32)^x
\][/tex]
Hence, the model that approximates the number of wasps in the colony for any given week [tex]\( x \)[/tex] is:
[tex]\[
4(1.32)^x
\][/tex]
Among the given options, the correct model is:
[tex]\[
4(1.32)^x
\][/tex]
