Chapter 0, Problem 65RE

i.

To determine

To find: Write the number of guppies as a function of time $t$ .

Answer to Problem 65RE

Number of guppies as a function of time $t$ is represented by $f\left(t\right)=4\times 2^t$ .

Explanation of Solution

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

Concept Used:

If an amount is increasing by a factor each time, then the function is known as exponential function. Mathematically represented by $f(t)=x\times y^t$ here, $x\text{is an intial amount}$ , $y\text{is an growth amount}$ and $t$ is the time.

Calculation:

The number of guppies in Susan’s aquarium doubles every day.

So, Number of guppies as a function of time $t$ is represented by $f\left(t\right)=4\times 2^t$ .

Conclusion:

Number of guppies as a function of time $t$ is represented by $f\left(t\right)=4\times 2^t$ .

ii.

To determine

To find: the number of guppies after 4 days and after 1 week.

Answer to Problem 65RE

Number of guppies after 4 days and 1 week are 64 and 512.

Explanation of Solution

Given: Number of guppies as function of time is $f\left(t\right)=4\times 2^t$

Concept Used:

Law of exponent: $a^n=a\times a\times a\times a\times \mathrm{..........}n\text{ times}$

  $\text{1 week = 7 days}$

Calculation:

the number of guppies after 4 days: as $t=4$

  $\begin{array}{l}f\left(4\right)=4\times 2^4\\ f\left(4\right)=4\times 16\\ f\left(4\right)=64\end{array}$

Now,

the number of guppies after 1 week: as $t=7$

  $\begin{array}{l}f\left(7\right)=4\times 2^7\\ f\left(7\right)=4\times 128\\ f\left(7\right)=512\end{array}$

Conclusion:

Number of guppies after 4 days and 1 week are 64 and 512.

iii.

To determine

To find: when will be there 2000 guppies.

Answer to Problem 65RE

It will take about 9 days to reach 2000 guppies.

Explanation of Solution

Given: Number of guppies are 2000.

Concept Used:

Properties of logarithm: $\ln a^m=m\times \ln a$

Calculation:

Number of guppies as function of time is $f\left(t\right)=4\times 2^t$

So,

  $\begin{array}{c}2000=4\times 2^t\\ \frac{2000}{4}=2^t\\ 500=2^t\end{array}$

Take log both side:

  $\begin{array}{c}\text{ln500=ln}{2}^t\\ \ln 500=t\times \ln 2\\ t=\frac{\ln 500}{\ln 2}\\ t=8.966\\ t\approx 9\end{array}$

Conclusion:

It will take about 9 days to reach 2000 guppies.

iv.

To determine

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

Answer to Problem 65RE

This model does not fit as there is a limitation on the number of guppies and their oxygen usage.

Explanation of Solution

Given: model of number of guppies is $f\left(t\right)=4\times 2^t$

Concept Used:

Concept of exponential function and its limitation.

Calculation:

This model doses not fit for Susan’s guppy population as there is a limitation to the number of guppies that can fit into the tank and the amount of oxygen they are going to use.

Conclusion:

This model does not fit as there is a limitation on the number of guppies and their oxygen usage.