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Illegal Fish
Task
A ﬁsherman illegally introduces some ﬁsh 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 “weekly percent growth rate” 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 ﬁsherman illegally introduces some ﬁsh 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?
Comments
Log in to commentKevin says:
6 monthsNice task; it seems we could get a bit more depth out of this task with two more questions:
" d. What is the percentage growth rate per week in question (b)?
e. Look for a pattern in questions (c) and (d). Use it to quickly find the percentage growth rate for the following values of b: (i) b = 1.15 (ii) b = 1.3 (iii) b = 1.7 (iv) b = 1 + r "
The "look for a pattern" prompt may or may not be desirable. In any case I think these questions might encourage students to see the structure of exponential growth functions.