Chapter 1, Problem 65RE

a.

To determine

To write: the number of guppies as a function of time t.

Answer to Problem 65RE

  $f(t)=4\cdot 2^t$ .

Explanation of Solution

Given information: The number of guppies in Susan's aquarium doubles every day. There are four guppies initially.

Calculation:

If an amount increases by a factor each time, the function is exponential so its of the form $f(t)=a.b^t$ , where a is the initial amount, b is the growth factor, and t is time.

Since the number of guppies is doubling, so b =2 and since 4 is the initial number of guppies, a = 4 .The equation is then $f(t)=4\cdot 2^t$ .

b.

To determine

To find: How many guppies were present after 4 days, after 1 week.

Answer to Problem 65RE

  $\begin{array}{c}4\text{days:}64\text{guppies}.\\ 1\text{week:}512\text{guppies}\end{array}$

Explanation of Solution

Given information: $f(t)=4\cdot 2^t$

Calculation:

  $\begin{array}{c}f(t)=4\cdot 2^t\\ t=4\text{days}\Rightarrow f(4)=4\cdot 2^4=64\text{guppies}.\\ 1\text{week}=7\text{days}\Rightarrow f(7)=4\cdot 2^7=512\text{guppies}\text{.}\end{array}$

c.

To determine

To find: when will there be 2000 guppies.

Answer to Problem 65RE

About 9 days.

Explanation of Solution

Given information: $f(t)=4\cdot 2^t$ .

Calculation:

  $\begin{array}{c}f(t)=4\cdot 2^t\\ f(t)=2000\\ 2000=4\cdot 2^t\\ 2^t=500\\ \ln 2^t=\ln (500)\\ t\ln (2)=\ln (500)\\ t=\frac{\ln (500}{\ln (2)}\approx 8.966\end{array}$

Since t is almost 9, it will take about 9 days.

d.

To determine

To give: reasons why this might not be a good model for the growth of Susan's guppy population.

Answer to Problem 65RE

Explanation of Solution

Given information: $f(t)=4\cdot 2^t$

Calculation:

Like any population, there has to be a maximum or carrying capacity due to the limitations of the environment. In this case, there is going to be a limit to the number of guppies that can fit into the tank and the amount of oxygen for their use.