If a population is declining at 12% per year and the current population size is 300 animals, how many animals will be in the population after 7 years?
Understand the Problem
The question describes a population of animals that is declining at a constant percentage rate each year. Given an initial population size, the question asks us to predict the population size after a certain number of years, taking into account the annual decline rate. We can use the exponential decay formula to solve this.
Answer
123
Answer for screen readers
123
Steps to Solve
- Identify the initial population, decay rate, and time period
The initial population $P_0$ is 300 animals. The decay rate $r$ is 12% per year, which is 0.12 as a decimal. The time period $t$ is 7 years.
- Apply the exponential decay formula
The formula for exponential decay is $P(t) = P_0(1 - r)^t$, where: $P(t)$ is the population after time $t$ $P_0$ is the initial population $r$ is the decay rate (as a decimal) $t$ is the time in years.
- Substitute the values into the formula
Substitute $P_0 = 300$, $r = 0.12$, and $t = 7$ into the formula: $P(7) = 300(1 - 0.12)^7$
- Simplify the expression
$P(7) = 300(0.88)^7$ $P(7) = 300(0.408622476)$
- Calculate the final population
$P(7) = 122.5867428$
- Round to the nearest whole number
Since we are talking about a population of animals, we should round to the nearest whole number. $P(7) \approx 123$
123
More Information
After 7 years, the population will be approximately 123 animals. This is based on a consistent 12% decline each year from the initial population of 300.
Tips
A common mistake is to use the growth formula instead of the decay formula, or forgetting to convert the percentage to a decimal. Also, some might not round to the nearest whole number, which is appropriate for this context. Forgetting to exponentiate the decay rate is also a common error.
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