Question

The bacteria in a 4-liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container?

Short answer

59 minutes.

Step-by-step solution

Understand the doubling behavior

The bacteria double in number every minute. This means that one minute before the container is full, it was half full.

Full container time

The container is full at 60 minutes.

Determine the time at half full

Since the bacteria double every minute, it must have taken only 59 minutes to fill half the container. One more minute of doubling would fill it completely.

Key concepts

Doubling Time

Doubling time is the period it takes for a quantity to double in size or value. In this exercise, the bacteria's population in the container doubles every minute. This means that if you start with any amount of bacteria, their number will be twice as large after one minute. Doubling time is significant because it allows us to predict how quickly something can grow. For mathematical representation, if you have an initial quantity of bacteria (let's call it N), after one doubling time (T), the amount will be 2N.

Hence, knowing the container is full at 60 minutes, it implies the bacteria count in the container at 59 minutes was half-full. This is the essence of understanding the doubling time. It simplifies many exponential growth calculations.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are often used to model growth processes like population or financial interests. In our exercise, the bacterial growth can be modeled by an exponential function due to its doubling nature. The general form of an exponential function is:

\[ f(t) = N \times e^{rt} \]

Here, 'N' is the initial amount, 'e' is the base of the natural logarithm, 'r' is the growth rate, and 't' is time.

For doubling specifically, the formula simplifies since the population doubles every minute. This can be shown as:
\[ P(t) = P_0 \times 2^t \]

In the context of our problem:
\[ P(59) = P_0 \times 2^{59} \] will be half of \[ P(60) = P_0 \times 2^{60} \].

This visualizes how exponential functions dictate the growth pattern of the bacterial population, and why the container is half full at 59 minutes when it's completely full at 60 minutes.

Bacterial Growth

Bacterial growth often follows an exponential trend, meaning the population of bacteria can grow very rapidly over a short period. In the scenario given, the bacteria double every minute. Initially, there might be just a few bacteria, but their numbers increase exponentially with time.

This kind of growth can be studied in a lab setting by placing bacteria in a nutrient-rich environment and observing their population growth over time.

• S-shaped growth curve: While our exercise focuses on the exponential phase, it is worth noting that real bacterial growth often eventually slows due to constraints like nutrient depletion or waste buildup, leading to an S-shaped or logistic growth curve.
• Importance in real-world applications: Understanding bacterial growth is crucial in fields like medicine, where controlling the spread of bacteria can be a matter of life and death.

By learning this concept, students can apply similar principles to other rapidly growing populations or entities, such as viral spread, cancer cells, or even financial investments.