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In a certain geographic location, a herd of antelopes is declining at a rate of six percent every year. If there are
currently 570 antelopes in the herd, create an exponential decay function to model the problem. If the decrease
in population continues at this rate, how many antelopes will be in the herd after 10 years? Round your answer
to the nearest whole number. (1 point)

Verified Answer :

GinnyAnswer

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:
\(\[ P(t) = P_0 \times (1 - r)^t \]\)
where:
- \(\( P(t) \)\) is the population at time \(\( t \)\)
- \(\( P_0 \)\) is the initial population
- \(\( r \)\) is the decline rate per period
- \(\( t \)\) is the number of periods (years, in this case)

Substituting the given values into the formula:
- \(\( P_0 = 570 \)\)
- \(\( r = 0.06 \)\)
- \(\( t = 10 \)\)

The equation becomes:
\(\[ P(10) = 570 \times (1 - 0.06)^{10} \]\)

Simplifying the term inside the parentheses:
\(\[ 1 - 0.06 = 0.94 \]\)

So the equation simplifies to:
\(\[ P(10) = 570 \times 0.94^{10} \]\)

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 \(\( 307 \)\).