Decode P(t)=40(1.3)^t: Population Growth Secrets Revealed

Unpacking the Mystery of Exponential Growth

Hey there, maths explorers! Today, we're diving deep into a super common, yet often misunderstood, mathematical concept: exponential growth. We've all seen functions like $P(t)=40(1.3)^t$ pop up in textbooks or online, representing everything from population booms to bacterial colonies, or even how your investments might grow over time. But what do these numbers actually mean? How do we translate this cool mathematical formula into something we can understand in plain English? That's what we're here to figure out, guys!

Our main mission today is to crack the code of the function $P(t)=40(1.3)^t$, which, as the initial prompt tells us, represents the size of a population. This function is a classic example of an exponential function, a mathematical powerhouse that describes situations where a quantity increases (or decreases) at a rate proportional to its current size. Think about it: a small population might add a few members, but a much larger population will add many more members in the same time frame, simply because there are more individuals to reproduce. That's the essence of exponential growth right there! It's not just adding the same amount each time; it's adding a percentage of what's already there, leading to often dramatic increases over time. This makes understanding its components absolutely crucial, not just for passing your math class, but for grasping how the world around us changes. We're going to break down each part of this formula so clearly that you'll feel like a maths wizard by the end. You'll learn how to identify the initial value, the growth factor, and most importantly, the percentage growth rate that's hidden within that simple-looking equation. We'll also address a specific question: which statement correctly describes this function? By the time we're done, you'll know exactly why one particular answer is the undisputed champion. So, buckle up, because we're about to demystify one of mathematics' most fascinating and powerful tools, giving you the skills to interpret similar growth models in any real-world scenario you might encounter. Let's get to it and unlock the secrets of $P(t)=40(1.3)^t$!

The Basics: What Do Those Numbers Really Mean?

Alright, let's get down to the nitty-gritty of $P(t)=40(1.3)^t$. When you see an exponential function describing growth, it generally follows a standard form: $P(t) = A(b)^t$. Each letter in this formula has a very specific and important job, and understanding them is key to decoding any exponential growth scenario. So, let's break down our specific function, $P(t)=40(1.3)^t$, piece by piece.

First up, we have $A$, which in our case is 40. What does this number represent? It's the initial amount or initial population at time $t=0$. Imagine hitting a stopwatch the moment you start observing something. At that exact initial moment ($t=0$), the population size is 40. Whether that's 40 bacteria, 40 rabbits, or 40 people, it's the starting point of our observation. So, if you were asked, "What was the population size at the very beginning of the study?" your answer would be a confident 40. This initial value is fundamental, as all subsequent growth is built upon this foundation. It sets the baseline from which all increases are calculated, making it a critical component for understanding the overall trajectory of the population.

Next, and perhaps the most crucial part for today's question, is $b$, which in our function is 1.3. This b is what we call the growth factor. The growth factor tells us how much the quantity multiplies by in each unit of time. It's a multiplier, not an adder. For growth, this b value will always be greater than 1. If it were less than 1 (but still positive), we'd be talking about decay! Now, here's where the magic happens and where many people get a little tripped up. The growth factor, b, is directly related to the growth rate, often represented as r. The relationship is simple: $b = 1 + r$. To find the actual percentage growth, we need to isolate r.

Let's do the math for our function $P(t)=40(1.3)^t$:

• We know $b = 1.3$.
• Using the formula $b = 1 + r$, we substitute: $1.3 = 1 + r$.
• To find r, we just subtract 1 from both sides: $r = 1.3 - 1$.
• This gives us $r = 0.3$.

Now, r is a decimal, and to express it as a percentage, we simply multiply by 100. So, $0.3 imes 100 = 30\%$. Aha! This means that for every unit of time (be it an hour, a day, a year – whatever t represents), the population grows by a staggering 30% of its current size. This is the critical insight we need to answer our initial question. The growth factor of 1.3 doesn't mean it grows by 1.3 times the original amount each period, but rather that it becomes 1.3 times itself from the previous period, which is equivalent to adding 30% to its prior value. Understanding this distinction between the growth factor and the growth rate is absolutely paramount. It’s what differentiates a true understanding of exponential functions from a superficial glance. The growth factor (1.3) signifies that each period the population is multiplied by 1.3. This multiplication inherently means that the population retains its previous 100% (the '1' in 1.3) and adds an extra 30% (the '.3' in 1.3) on top of that. This simple calculation of $b-1$ is the golden ticket to unlocking the true percentage growth rate concealed within any exponential growth function, making it easy to see why 30% is the correct interpretation.

Beyond the Numbers: Visualizing Population Growth Over Time

Okay, guys, we've nailed down what the numbers mean individually. Now, let's explore what happens when we let time, t, start ticking! The real power of an exponential function like $P(t)=40(1.3)^t$ isn't just in its initial state or its growth rate, but in how it describes change over multiple time periods. This is where the magic (and sometimes the shock) of exponential growth truly comes alive. It's not just a steady, linear increase; it's a rapidly accelerating upward curve that can lead to mind-boggling numbers surprisingly quickly.

Let's put some numbers into our function to see this in action:

• At $t=0$ (the start): $P(0) = 40(1.3)^0 = 40 imes 1 = 40$. As we already discussed, this is our initial population. It's our baseline, our starting point, before any growth has occurred. Easy peasy, right?

• At $t=1$ (one time unit later): $P(1) = 40(1.3)^1 = 40 imes 1.3 = 52$. So, after the first period, our population has grown from 40 to 52. Let's confirm our 30% growth: 30% of 40 is $0.30 imes 40 = 12$. Adding this to the initial population: $40 + 12 = 52$. Bingo! The 30% growth is clearly visible here. This demonstrates how the growth factor of 1.3 directly translates to a 30% increase; the population is now 130% of its initial size.

• At $t=2$ (two time units later): $P(2) = 40(1.3)^2 = 40 imes (1.3 imes 1.3) = 40 imes 1.69 = 67.6$. Notice something crucial here? The growth for the second period isn't 12, like the first period. Instead, it's 30% of the new population size (52), which is $0.30 imes 52 = 15.6$. Add that to 52: $52 + 15.6 = 67.6$. This is the defining characteristic of exponential growth: the amount of increase gets larger and larger with each passing time unit, even though the percentage rate of growth (30%) remains constant. The population isn't just adding a fixed number of individuals; it's adding 30% of whatever its current size is, which means the actual count of new individuals accelerates over time. This dynamic is what makes exponential functions so powerful and, frankly, sometimes a bit scary in real-world contexts, because the increases can quickly outpace our intuitive expectations. It's truly a snowball effect, where the snow at the beginning might be small, but the snow added gets bigger and bigger as the snowball rolls. This multiplicative nature stands in stark contrast to linear growth, where a fixed amount is added each period, resulting in a straight-line graph. Exponential growth, with its ever-steepening curve, can lead to astonishingly large numbers, highlighting its critical role in modelling everything from financial investments that compound over time to the rapid spread of information or even diseases. Understanding this visual and numerical progression is key to fully appreciating the function's implications, showing us that the impact of a constant percentage growth rate becomes exponentially greater as the base population itself grows. It’s an incredibly important concept that helps us predict and understand long-term trends in many different fields.

Decoding the Options: Why C is the Clear Winner!

Alright, guys, this is where all our hard work pays off! We've dissected the function $P(t)=40(1.3)^t$ and figured out what each piece means. Now, let's look at the given options and see which one correctly describes our population growth. This is the moment of truth, and you'll see why understanding the difference between the initial value, the growth factor, and the growth rate is absolutely essential for nailing questions like these.

Let's evaluate each statement one by one:

• A. Growth by 40%: At first glance, you might see that 40 in the function and think,