Bacterial Growth Unveiled Calculating Initial Size Doubling Period And Population After 95 Minutes

Hey everyone! Today, we're diving into a fascinating problem involving exponential growth, specifically in a bacteria culture. This is a classic example that pops up in biology, mathematics, and even finance, so understanding it is super valuable. We've got a scenario where a bacteria culture starts with an unknown number of bacteria, and its population grows exponentially over time. We're given two data points: the population after 15 minutes and the population after 30 minutes. Our mission? To find the initial size of the culture, the doubling period, and the population after a specific time (95 minutes).

Unpacking the Problem

Before we jump into the math, let's break down what we're dealing with. Exponential growth is a phenomenon where a quantity increases at a rate proportional to its current value. Think of it like this: the more bacteria you have, the faster they reproduce, leading to an accelerating growth rate. This is different from linear growth, where the quantity increases at a constant rate. In the context of our bacteria culture, this means that the population isn't just increasing by a fixed number of bacteria every minute; instead, the number of new bacteria added each minute grows as the population grows.

The problem gives us two key pieces of information:

• At 15 minutes, the population is 100.
• At 30 minutes, the population is 1400.

These are our anchor points. We'll use them to build our mathematical model and answer the questions. The questions we need to answer are:

1. What was the initial size of the culture?
2. What is the doubling period?
3. What is the population after 95 minutes?

Setting Up the Exponential Growth Model

The heart of solving this problem lies in understanding the formula for exponential growth. The general form of an exponential growth equation is:

N(t) = N₀ * e^(kt)

Where:

• N(t) is the population at time t
• N₀ is the initial population (what we're trying to find in the first question)
• e is the base of the natural logarithm (approximately 2.71828)
• k is the growth constant (a positive number that determines how quickly the population grows)
• t is the time

This formula basically says that the population at any time t is equal to the initial population N₀ multiplied by e raised to the power of kt. The growth constant k is crucial because it dictates the rate of growth. A larger k means faster growth, while a smaller k means slower growth.

Our first task is to find N₀ and k. We have two data points, which means we can create two equations using the exponential growth formula:

1. 100 = N₀ * e^(15k) (Population at 15 minutes)
2. 1400 = N₀ * e^(30k) (Population at 30 minutes)

Now we have a system of two equations with two unknowns (N₀ and k). There are a couple of ways to solve this system, but a common and efficient method is to divide the second equation by the first equation. This will eliminate N₀ and allow us to solve for k.

Solving for the Growth Constant (k)

Dividing the second equation by the first, we get:

1400 / 100 = (N₀ * e^(30k)) / (N₀ * e^(15k))

Simplifying, we get:

14 = e^(30k - 15k)
14 = e^(15k)

To isolate k, we take the natural logarithm (ln) of both sides:

ln(14) = ln(e^(15k))
ln(14) = 15k

Now, we solve for k:

k = ln(14) / 15
k ≈ 0.17976

So, the growth constant k is approximately 0.17976. This value tells us how rapidly the bacteria population is increasing. Remember this value, as we'll need it later to calculate the doubling period and the population at 95 minutes.

Finding the Initial Population (N₀)

Now that we have k, we can plug it back into either of our original equations to solve for N₀. Let's use the first equation:

100 = N₀ * e^(15 * 0.17976)

Simplifying:

100 = N₀ * e^(2.6964)
100 = N₀ * 14

Solving for N₀:

N₀ = 100 / 14
N₀ ≈ 7.14

Therefore, the initial size of the culture was approximately 7.14 bacteria. Since we can't have fractions of bacteria in a real-world scenario, we might round this to 7. However, for the sake of accuracy in further calculations, we'll keep the more precise value.

Calculating the Doubling Period

The doubling period is the time it takes for the population to double in size. This is a crucial metric in understanding exponential growth. To find the doubling period, we need to figure out how long it takes for the population to become 2 * N₀. Let's call the doubling period T. We can set up the following equation:

2 * N₀ = N₀ * e^(kT)

Notice that N₀ cancels out:

2 = e^(kT)

Now we take the natural logarithm of both sides:

ln(2) = ln(e^(kT))
ln(2) = kT

Solving for T:

T = ln(2) / k

We already know k is approximately 0.17976, so:

T = ln(2) / 0.17976
T ≈ 3.856 minutes

So, the doubling period for this bacteria culture is approximately 3.856 minutes. This means that the population doubles roughly every 3.856 minutes, highlighting the rapid nature of exponential growth. This is a crucial piece of information for understanding the dynamics of the bacteria culture.

Predicting the Population After 95 Minutes

Now, let's tackle the final question: What is the population after 95 minutes? We have all the pieces we need: the initial population N₀, the growth constant k, and the time t. We simply plug these values into our exponential growth formula:

N(t) = N₀ * e^(kt)
N(95) = 7.14 * e^(0.17976 * 95)

Simplifying:

N(95) = 7.14 * e^(17.0772)
N(95) ≈ 7.14 * 26124657.6
N(95) ≈ 186518145.8

Therefore, the population after 95 minutes is approximately 186,518,145.8 bacteria. Wow, that's a massive number! This really drives home the power of exponential growth. Starting from just a few bacteria, the population explodes to over 186 million in just 95 minutes.

Key Takeaways and Real-World Applications

Let's recap what we've learned:

• We used the exponential growth formula N(t) = N₀ * e^(kt) to model the growth of a bacteria culture.
• We solved for the initial population N₀ (approximately 7.14 bacteria) and the growth constant k (approximately 0.17976) using the given data points.
• We calculated the doubling period (approximately 3.856 minutes), which tells us how quickly the population doubles.
• We predicted the population after 95 minutes (approximately 186,518,145.8 bacteria), demonstrating the rapid increase characteristic of exponential growth.

This type of problem isn't just confined to bacteria cultures. Exponential growth models are used in various fields, including:

• Finance: To model compound interest and investment growth.
• Biology: To study population growth, disease spread, and drug decay.
• Physics: To describe radioactive decay.
• Computer Science: To analyze the efficiency of algorithms.

Understanding exponential growth is a crucial skill in many disciplines. The ability to model and predict exponential growth allows us to make informed decisions and understand the world around us better.

Final Thoughts

This problem might seem complex at first, but by breaking it down step-by-step, we can see how the exponential growth formula and basic algebra can be used to solve it. The key is to understand the underlying principles of exponential growth and how to apply the formula correctly. So, the next time you encounter a scenario involving exponential growth, remember the lessons we've learned today! You've got this!