Sasquatch Population Below 75: When Was It?

Hey guys! Let's dive into a fascinating mathematical problem concerning a hypothetical Sasquatch population in Salt Lake County. We'll be using a given function to figure out during which years the population was fewer than 75 individuals. This is a cool example of how math can be used to model and understand population trends, even for the most elusive of creatures! So, buckle up, and let's get started!

The Sasquatch Population Model

At the heart of our investigation is the population model: P(t) = 450t / (t + 45). This equation, my friends, is a mathematical representation of how the Sasquatch population, P(t), changes over time, t. It's a rational function, which is often used to model situations where growth slows down as it approaches a carrying capacity (in this case, a maximum sustainable population). It is crucial to understand that in this model, t = 0 represents the year 1803. This is our starting point, our baseline for measuring time in this Sasquatch population study. We're tasked with finding the years when the population, P(t), was less than 75. This means we need to solve an inequality, a mathematical statement that compares two expressions using symbols like "less than" (<) or "greater than" (>). Solving this inequality will give us the range of t values, which we can then convert into years to answer our question. This is where the fun begins, where we get to put our math skills to the test and unravel the mysteries of this Sasquatch population. Remember, the beauty of mathematical models lies in their ability to simplify complex real-world scenarios, allowing us to analyze and make predictions. This model, while hypothetical, gives us a glimpse into how population dynamics can be studied using mathematical tools. So, let's roll up our sleeves and get to work on solving this intriguing problem!

Setting Up the Inequality

Alright, let's translate our problem into a mathematical inequality. We're looking for the times when the Sasquatch population, P(t), is less than 75. Mathematically, this is expressed as: 450t / (t + 45) < 75. This inequality is the key to unlocking the answer. It's a concise way of stating our research question, and it provides the framework for our mathematical manipulations. Now, to solve this inequality, we need to isolate t on one side. This involves a series of algebraic steps, each carefully designed to maintain the integrity of the inequality. We'll be multiplying, dividing, and simplifying, all while making sure that the direction of the inequality sign remains correct. Solving inequalities is a bit like solving equations, but there are a few extra things to keep in mind. For example, multiplying or dividing both sides by a negative number flips the direction of the inequality sign. This is a crucial detail that we'll need to pay attention to as we work through the problem. But don't worry, we'll take it one step at a time, and I'll guide you through each step. The goal here is not just to get the right answer, but also to understand the process, the logic behind each operation. Because once you understand the process, you can apply it to other similar problems, and that's where the real learning happens. So, let's tackle this inequality and uncover the secrets of the Sasquatch population!

Solving the Inequality

Okay, guys, let's get our hands dirty and solve this inequality: 450t / (t + 45) < 75. The first step is to get rid of the fraction. We can do this by multiplying both sides of the inequality by (t + 45). But hold on! We need to be careful here. Since t represents time, it's always going to be greater than or equal to 0. This means (t + 45) will always be positive. Why is this important? Because multiplying or dividing an inequality by a positive number doesn't change the direction of the inequality sign. Phew! So, we can safely multiply both sides by (t + 45) without flipping the sign. This gives us: 450t < 75(t + 45). Now, let's distribute the 75 on the right side: 450t < 75t + 3375. Next, we want to get all the t terms on one side. So, let's subtract 75t from both sides: 375t < 3375. Finally, to isolate t, we divide both sides by 375: t < 9. There we have it! Our solution to the inequality is t < 9. This means that the Sasquatch population was less than 75 for times t less than 9. But remember, t is in years since 1803. So, we need to translate this back into actual years to answer our original question.

Interpreting the Result

Fantastic work, everyone! We've solved the inequality and found that t < 9. But what does this actually mean in terms of years? Remember, t = 0 represents the year 1803. So, t = 9 represents 9 years after 1803, which is the year 1812. Our inequality, t < 9, tells us that the Sasquatch population was less than 75 before the year 1812. To be precise, it was less than 75 from the year 1803 (when t = 0) up until the year 1812. So, to answer the original question,