Illegal Fish
Task
A ï¬sherman illegally introduces some ï¬sh into a lake, and they quickly propagate. The growth of the population of this new species (within a period of a few years) is modeled by $P(x)=5b^x$, where $x$ is the time in weeks following the introduction and $b$ is a positive unknown base.

Exactly how many fish did the fisherman release into the lake?

Find $b$ if you know the lake contains 33 fish after eight weeks. Show stepbystep work.

Instead, now suppose that $P(x)=5b^x$ and $b=2$. What is the weekly percent growth rate in this case? What does this mean in everyday language?
IM Commentary
This is a direct task suitable for the early stages of learning about exponential functions. Students interpret the relevant parameters in terms of the realworld context and describe exponential growth.
Solution

The fisherman released the fish into the lake at time zero, $t=0$, the exact moment of introduction. Thus, the number of fish that the fisherman released into the lake is given by:
$$ \begin{align} P(0) &= 5b^0 \\ P(0) &= 5 \cdot 1 \\ P(0) &= 5 \end{align} $$This means that the fisherman released 5 fish into the lake.

We know that $x$ is the time in weeks following the introduction. Let us assume that 2 months is approximately 8 weeks, giving $t=8$. Then, if the lake contains 33 fish after two months, or $P(8)=33$, we can solve for $b$:
$$ \begin{align} 33 &= 5b^8 \\ b^8 &= \frac{33}{5} \\ b &= \left( \frac{33}{5} \right)^{\frac18} \\ b &\approx 1.266 \end{align} $$Thus, $b$ is approximately equal to 1.2 if the lake contains 33 fish after two months.
The â€œweekly percent growth rateâ€ is the percent increase of the population in one week. Since $b=2$, we know that the population at any time $x$ is given by $P(x)=5\cdot 2^x$, and that the population one week later is given by $$ P(x+1)=5\cdot 2^{x+1}=(5\cdot 2^x)\cdot 2=2P(x). $$ We learn that the population doubles each week, which is to say that there is a 100% weekly growth rate.
Illegal Fish
A ï¬sherman illegally introduces some ï¬sh into a lake, and they quickly propagate. The growth of the population of this new species (within a period of a few years) is modeled by $P(x)=5b^x$, where $x$ is the time in weeks following the introduction and $b$ is a positive unknown base.

Exactly how many fish did the fisherman release into the lake?

Find $b$ if you know the lake contains 33 fish after eight weeks. Show stepbystep work.

Instead, now suppose that $P(x)=5b^x$ and $b=2$. What is the weekly percent growth rate in this case? What does this mean in everyday language?