Write an exponential equation in the form y = a(b)^x that can model the population of collared pikas on Little Oak Mountain, y, in x years.
Understand the Problem
The question is asking us to formulate an exponential equation to model the decreasing population of collared pikas on Little Oak Mountain over time. We need to identify the initial population and the decay factor based on the given rate of decrease.
Answer
The exponential equation for the population of collared pikas is: $$ y = 750 \left(\frac{4}{5}\right)^x $$
Answer for screen readers
The exponential equation modeling the population of collared pikas is given by:
$$ y = 750 \left(\frac{4}{5}\right)^x $$
Steps to Solve
- Identify Initial Population
The initial population of collared pikas is given as 750. Thus, we set $a = 750$.
- Determine the Decay Factor
The population decreases by $ \frac{1}{5} $. This means that $ \frac{4}{5} $ of the population remains each year. Therefore, the decay factor is $b = \frac{4}{5}$.
- Formulate the Exponential Equation
Using the formula $y = a(b)^x$, we substitute the values we've identified:
$$ y = 750 \left(\frac{4}{5}\right)^x $$
This is the exponential model for the population decline of collared pikas over time.
The exponential equation modeling the population of collared pikas is given by:
$$ y = 750 \left(\frac{4}{5}\right)^x $$
More Information
This model shows how the population of collared pikas is expected to decline over time. Each year, approximately 80% of the population continues to survive, reflecting the significant impact of the annual decrease.
Tips
- Confusing the amount of decrease with the remaining population. It's essential to recognize that a decrease of ( \frac{1}{5} ) means that ( \frac{4}{5} ) is remaining.
- Incorrectly applying the decay factor. The decay factor must reflect the portion that remains after the decrease.
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