Chapter 3.8, Problem 23E

To determine

To find: the expression for the number of bacteria after $t$, and estimate the rate of growth of the bacteria population after 2.5 hours.

Answer to Problem 23E

Expression for growth of $n$ bacteria in $t$ hours is $n\left(t\right)=400\times 3^t$.

The rate of growth of bacteria after 2.5 hours is $6,850\text{units}/\text{hour}$.

Explanation of Solution

Given:

The initial population $n_0$ is 400.

Population of the bacteria triples in one hour as per the rate expression given below.

$n\left(t\right)=n_0\times 3^t$ (1)

Calculation:

Expression for $n$ bacteria in $t$ hours.

Substitute the 400 for $n_0$ in (2).

$n\left(t\right)=400\times 3^t$ (2)

Determine the rate of growth of population after 2.5 hours.

Differentiate equation (2) with respect to $t$.

$\begin{array}{c}\frac{dn}{dt}=400\times 3^t\left(\ln 3\right)\\ =400\times 3^t\left(1.098\right)\end{array}$

$\frac{dn}{dt}=439.44\times 3^t$ (3)

Substitute 2.5 for $t$ in equation (4).

$\begin{array}{c}\frac{dn}{dt}=439.44\times 3^{2.5}\\ =6,850.19\\ \frac{dn}{dt}\simeq 6,850\end{array}$

Expression for growth of $n$ bacteria in $t$ hours is $n\left(t\right)=400\times 3^t$.

The rate of growth of bacteria after 2.5 hours is $6,850\text{units}/\text{hour}$.