Question

A bacterial culture starts with 2000 bacteria and doubles every \(0.75 \mathrm{h}\). After how many hours will the bacteria count be 32 000?

Short answer

It takes 3 hours for the bacteria count to reach 32,000.

Step-by-step solution

Understand the problem

The problem involves exponential growth where the bacterial culture starts with 2000 bacteria and doubles every 0.75 hours. We need to find the time it takes for the bacteria count to reach 32,000.

Define the exponential growth formula

The formula for exponential growth is given by \[ N(t) = N_0 \times 2^{(t/T)} \]where:- \(N(t)\) is the number of bacteria at time \(t\)- \(N_0\) is the initial number of bacteria- \(T\) is the doubling time- \(t\) is the time in hours.

Substitute the known values into the formula

Given:\( N_0 = 2000 \)\( T = 0.75 \) hours \( N(t) = 32000 \)Substitute these values into the formula:\[ 32000 = 2000 \times 2^{(t/0.75)} \]

Solve for \(t\)

First, divide both sides by 2000 to isolate the exponential term:\[ \frac{32000}{2000} = 2^{(t/0.75)} \]\[ 16 = 2^{(t/0.75)} \]Next, express 16 as a power of 2:\[ 16 = 2^4 \]So we have:\[ 2^4 = 2^{(t/0.75)} \]Since the bases are the same, we can set the exponents equal to each other:\[ 4 = \frac{t}{0.75} \]Multiply both sides by 0.75:\[ t = 4 \times 0.75 \]\[ t = 3 \]

Key concepts

Doubling Time

In the context of bacterial growth, doubling time is the period it takes for a population of bacteria to double in number. Knowing the doubling time helps predict how quickly a bacterial population can grow under ideal conditions. For example, if we start with 2000 bacteria and the doubling time is 0.75 hours, we know that in 0.75 hours, the bacteria count will be 4000, in another 0.75 hours it will be 8000, and so on. This predictable pattern allows us to calculate how long it will take to reach any given population size. Doubling time is a critical concept in understanding exponential growth.

Exponential Growth Formula

Exponential growth describes how a quantity increases rapidly in proportion to its current value. For bacterial growth, the exponential growth formula is: \[ N(t) = N_0 \times 2^{(t/T)} \] Where:

• \(N(t)\): the number of bacteria at time \(t\)
• \(N_0\): the initial number of bacteria
• \(T\): the doubling time
• \(t\): the time in hours

This formula helps us to understand and predict the growth of a bacterial population over time. By plugging in different values, we can determine how long it will take for the bacteria to reach a certain number.

Isolate Exponential Term

To solve exponential growth problems, it's often necessary to isolate the exponential term. In the given problem, the equation becomes:\[ 32000 = 2000 \times 2^{(t/0.75)} \]We start by dividing both sides by 2000:\[ \frac{32000}{2000} = 2^{(t/0.75)} \]This simplifies to:\[ 16 = 2^{(t/0.75)} \]Next, express 16 as a power of 2:\[ 16 = 2^4 \]Now, equate the exponents because the bases are the same:\[ 4 = \frac{t}{0.75} \]Finally, solve for \(t\) by multiplying both sides by 0.75:\[ t = 4 \times 0.75 \]This gives us \(t = 3\) hours. So, it takes 3 hours for the bacterial population to grow from 2000 to 32,000. Isolating the exponential term is crucial in solving these equations effectively.