To determine which equation represents a population of 250 animals that decreases at an annual rate of 21%, we need to understand how population change is modeled mathematically.
When dealing with exponential decay, the formula we use is:
[tex]\[ p = P_0 \cdot (1 - r)^t \][/tex]
where:
- [tex]\( p \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the annual decrease rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the number of years.
Given the problem:
- The initial population, [tex]\( P_0 \)[/tex], is 250 animals.
- The annual decrease rate, [tex]\( r \)[/tex], is 21%, which can be expressed as [tex]\( 0.21 \)[/tex].
Substituting these values into the formula, we get:
[tex]\[ p = 250 \cdot (1 - 0.21)^t \][/tex]
Simplifying inside the parentheses gives:
[tex]\[ p = 250 \cdot (0.79)^t \][/tex]
Thus, the correct equation that represents the population decreasing at an annual rate of 21% is:
[tex]\[ \boxed{p = 250(0.79)^t} \][/tex]
From the provided answer choices:
A. [tex]\( p=250(1.79)^t \)[/tex]: This represents a 79% increase, not a decrease.
B. [tex]\( p=250(0.79)^t \)[/tex]: Correct, since it represents a 21% annual decrease.
C. [tex]\( p=250(0.21)^t \)[/tex]: This represents the population retaining only 21%, which is incorrect.
D. [tex]\( p=250(1.21)^t \)[/tex]: This represents a 21% increase, not a decrease.
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
