Chapter 6.4, Problem 31E

To determine

To find: the error in given solution.

Answer to Problem 31E

The error is in the model for exponential growth function for the given data.

Correct function is $b\left(t\right)=10{\left(2.5\right)}^t$

  $b\left(8\right)\approx 15,259$

Explanation of Solution

Given:

The growth rate of bacterial culture is 150% each hour. Initially, there are 10 bacteria.

Then, number of bacteria in culture after 8 hours is:

  $\begin{array}{c}b\left(t\right)=10{\left(1.5\right)}^t\\ b\left(8\right)=10{\left(1.5\right)}^8\\ \approx 256.3\end{array}$

So, after 8 hours, there are 256 bacteria in the culture.

Concept used:

Exponential growth function:

  $y=a{\left(1+r\right)}^t$ ….. (1)

Here, the y is the final value, a is the initial value, and r is the rate of growth (in decimal form).

Calculation:

The error here is in the model for exponential growth function for the given data.

Here, $a=10$ and $r=150\%=1.5$

So, the correct function is

  $\begin{array}{c}b\left(t\right)=a{\left(1+r\right)}^t\\ =10{\left(1+1.5\right)}^t\\ =10{\left(2.5\right)}^t\end{array}$

Then, number of bacteria in culture after 8 hours is:

  $\begin{array}{c}b\left(8\right)=10{\left(2.5\right)}^8\\ \approx 10\left(1525.879\right)\\ \approx 15,258.79\end{array}$

So, number of bacteria in culture after 8 hours will be about 15,259.