Chapter 1.5, Problem 30E

A bacteria culture starts with 500 bacteria and doubles in size every half hour.

(a) How many bacteria are there after 3 hours?

(b) How many bacteria are there after t hours?

(c) How many bacteria are there after 40 minutes?

(d) Graph the population function and estimate the time for the population to reach 100,000.

To determine

To Find: The population of bacteria after 3 hours.

Answer to Problem 30E

There are 32,000 bacteria after 3 hours.

Explanation of Solution

Given:

The initial population of bacteria is 500 and it gets double for every half hour.

Calculation:

Using the given initial population of bacteria $P\left(0\right)$ and the rate of growth, define a function to calculate the population of bacteria in terms of time t as shown below,

$\begin{array}{c}P\left(t\right)=2^{\frac{t}{0.5}}\cdot P\left(0\right)\\ =2^{2t}\cdot P\left(0\right)\\ =4^t\cdot P\left(0\right)\end{array}$

Therefore, $P\left(t\right)=4^t\times P\left(0\right)$

Substitute $t=3$ to get the population of bacteria after 3 hours.

$\begin{array}{c}P\left(3\right)=2^{2\left(3\right)}\times 500\\ =2^6\times 500\\ =64\times 500\\ =32,000\end{array}$

Therefore, there are 32,000 bacteria after 3 hours.

To determine

To find: The population of bacteria after t hours.

Answer to Problem 30E

There are $4^t\times 500$ bacteria after t hours.

Explanation of Solution

Given:

The initial population of bacteria is 500 and it gets double for every half hour.

Calculation:

Using the given initial population of bacteria $P\left(0\right)$ and the rate of growth, define a function to calculate the population of bacteria in terms of time t as shown below,

$\begin{array}{c}P\left(t\right)=2^{\frac{t}{0.5}}\cdot P\left(0\right)\\ =2^{2t}\cdot P\left(0\right)\\ =4^t\cdot P\left(0\right)\end{array}$

Therefore, $P\left(t\right)=4^t\times P\left(0\right)$

Substitute $P\left(0\right)=500$ and obtain the expression for population of bacteria after t hours.

Therefore, there are $4^t\times 500$ bacteria after t hours.

To determine

To Find: The population of bacteria after 40 minutes.

Answer to Problem 30E

There are 1260 bacteria after 40 minutes.

Explanation of Solution

Given:

The initial population of bacteria is 500 and it gets double for every half hour.

Calculation:

Using the given initial population of bacteria $P\left(0\right)$ and the rate of growth, define a function to calculate the population of bacteria in terms of time t as shown below,

$\begin{array}{c}P\left(t\right)=2^{\frac{t}{0.5}}\cdot P\left(0\right)\\ =2^{2t}\cdot P\left(0\right)\\ =4^t\cdot P\left(0\right)\end{array}$

Therefore, $P\left(t\right)=4^t\times P\left(0\right)$

Convert the given time into hour as shown below,

40minutes = $\frac{40}{60}$ hours= $\frac{2}{3}$ hours.

Substitute $t=\frac{2}{3}$ to calculate the population of bacteria after $\frac{2}{3}$ hours follows,

$\begin{array}{c}P\left(\frac{2}{3}\right)=4^{\left(\frac{2}{3}\right)}\times 500\\ ={16}^{\frac{1}{3}}\times 500\\ =2.52\times 500\\ \approx 1260\end{array}$

Therefore,. there are 1260 bacteria after 40 minutes.

To determine

To estimate: The time for the population $P\left(t\right)$ to reach 100,000.

Explanation of Solution

Graph:

Use online graph calculator and draw the graph of the functions $P\left(t\right)=4^t\times 500$ as shown in the below Figure1.

From Figure 1, the curve intersects the line $y=100000$ at the point $\left(3.822,100000\right)$ and that means the time is 3.822 hours when the population becomes 100000.

That is, t=3.822 when $P\left(t\right)=100000$.

Therefore it takes 3.822 hours to reach to 100000 to the population.