To address this problem, we need to use the concept of exponential decay, which is often used to model situations where a quantity decreases at a constant percentage rate over time.
First, let's establish the given information:
- The current population of the antelopes: 570
- The annual decline rate: 6% (or 0.06)
- The number of years over which to calculate the decline: 10 years
The general formula for exponential decay is:
[tex]\[ P(t) = P_0 \times (1 - r)^t \][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( r \)[/tex] is the decline rate per period
- [tex]\( t \)[/tex] is the number of periods (years, in this case)
Substituting the given values into the formula:
- [tex]\( P_0 = 570 \)[/tex]
- [tex]\( r = 0.06 \)[/tex]
- [tex]\( t = 10 \)[/tex]
The equation becomes:
[tex]\[ P(10) = 570 \times (1 - 0.06)^{10} \][/tex]
Simplifying the term inside the parentheses:
[tex]\[ 1 - 0.06 = 0.94 \][/tex]
So the equation simplifies to:
[tex]\[ P(10) = 570 \times 0.94^{10} \][/tex]
By calculating the above expression, we find the population after 10 years.
After computing the value, the resulting population after 10 years is approximately 307 antelopes.
Thus, the population of the herd after 10 years, rounded to the nearest whole number, is [tex]\( 307 \)[/tex].
