Write an exponential equation in the form y = a(b)^x that can model the park's black bear population, y, in x decades.
Understand the Problem
The question is asking to write an exponential equation that models the black bear population in Pine Ridge State Park based on the current population and its growth rate. It states that there are currently 140 bears and that the population will increase by a factor of 1/5 each decade.
Answer
The exponential equation is $y = 140\left(\frac{6}{5}\right)^x$.
Answer for screen readers
The exponential equation that models the black bear population is:
$$ y = 140\left(\frac{6}{5}\right)^x $$
Steps to Solve
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Identify the initial population The initial population of black bears is given as 140. This will be the starting value in our exponential equation, denoted as $a$.
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Determine the growth factor The population increases by a factor of $\frac{1}{5}$ each decade. To express this in terms of growth, we need to convert this into a growth multiplier. The population after each decade will be multiplied by $1 + \frac{1}{5} = \frac{6}{5}$. So, $b = \frac{6}{5}$.
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Formulate the exponential equation With the initial population ($a = 140$) and the growth factor ($b = \frac{6}{5}$), we can write the model in the exponential form:
$$ y = a(b)^x $$
Substituting the values we have:
$$ y = 140\left(\frac{6}{5}\right)^x $$
The exponential equation that models the black bear population is:
$$ y = 140\left(\frac{6}{5}\right)^x $$
More Information
This equation means that for every decade that passes (as $x$ increases), the population of black bears will grow by a factor of $\frac{6}{5}$, or 1.2 times its size from the previous decade. Specifically, starting from 140 bears, the population will increase to 168 after one decade, then to 201.6 after two decades, and so on.
Tips
- Confusing increase with direct addition: Some may mistakenly add (1/5) to the population directly instead of applying the growth factor as (1 + \frac{1}{5}).
- Incorrect formulation of the exponent: Ensure to maintain the expression intact; it will be present in the format (b^x), making sure to replace (b) with the correct growth rate.
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