Time Interval For Stick Above 20 Feet: Explained!

Hey Plastik Magazine readers! Let's dive into a fun math problem today. We're going to figure out how long a stick stays above 20 feet in the air after being thrown straight up. Sounds cool, right? Let’s break it down step by step.

Understanding the Problem

In this mathematical scenario, we've got Tim tossing a stick straight up, and we're given a function that models the height of the stick over time. The height function is crucial here: h = -16t^2 + 48t. This equation tells us the height (h) of the stick in feet at any given time (t) in seconds. The goal is to find the time interval during which the stick is more than 20 feet above the ground. This isn’t just about plugging in numbers; it’s about understanding how the equation represents the real-world motion of the stick. This involves some algebra, but don't worry, we'll make it super clear.

First off, let's talk about what this equation actually means. The -16t^2 part? That's gravity doing its thing, pulling the stick back down. The +48t? That’s Tim's initial throw sending the stick upwards. Put them together, and you get a beautiful curve (a parabola, if you want to get technical) that shows the stick's journey through the air. What we're trying to find is a slice of time – a starting point and an ending point – where the stick's height is above that 20-foot mark. Think of it like drawing a horizontal line at 20 feet on the graph of this equation; we want to know the x-values (time) where the curve is above the line. This problem isn't just a bunch of numbers; it's a story about physics in action!

To really get our heads around this, let’s think about the stick’s trajectory. Tim throws it up, it goes up, up, up, slows down, reaches its peak, and then starts falling back down. We're not interested in the whole flight, just the part where it's higher than 20 feet. That means there are two points in time we care about: the moment it passes 20 feet on the way up, and the moment it drops back below 20 feet on the way down. Finding these two times is the key to solving the problem. Once we have those, we'll know exactly how long the stick was enjoying its time above the 20-foot mark. So, buckle up, because we're about to use some math to tell this stick's story!

Setting Up the Inequality

To find the time interval, we need to set up an inequality. We want to know when the height, h, is greater than 20 feet. So, we write: -16t^2 + 48t > 20. This inequality is the heart of our problem. It’s saying, “Hey, when is this function (the stick's height) bigger than 20?” Solving this will give us the times we're looking for. But before we jump into solving, let's talk about why we use an inequality here. It's not just about finding one specific time; it's about finding a range of times. The stick isn't just at 20 feet for a split second; it’s above 20 feet for a period of time. That's why we need that little “greater than” symbol.

This inequality is our roadmap. It takes the words of the problem and turns them into a mathematical statement we can actually work with. Think of it like translating a sentence from English into Math – you need to get all the pieces in the right order to make sense. In this case, we've taken the idea of the stick being “more than 20 feet high” and translated it into “-16t^2 + 48t > 20”. This step is crucial because it connects the real-world situation with the tools of algebra. Without this inequality, we'd just be staring at a bunch of numbers. Now, we have a mission: to solve it!

Before we start crunching numbers, let’s think about what the solution will look like. We're not expecting a single answer, like t = 2. We're expecting a range of times, something like “between 1 second and 2 seconds.” This makes sense if you picture the stick flying up and then coming down. It’ll be above 20 feet for a stretch of time, not just at one particular moment. Keeping this in mind will help us make sure our answer is reasonable when we finally get there. So, inequality set up, expectations set – let’s move on to the next step!

Solving the Inequality

Okay, guys, now comes the fun part: actually solving the inequality. First, we need to rearrange it a bit to make it easier to work with. Let's subtract 20 from both sides to get: -16t^2 + 48t - 20 > 0. This is a classic move in algebra – we want to get everything on one side and zero on the other. Why? Because now we can think about this as finding when a quadratic equation is greater than zero, which is a problem we know how to tackle.

Next up, to make things even smoother, let's divide the entire inequality by -4. But, and this is a big but, when we divide (or multiply) an inequality by a negative number, we have to flip the inequality sign. So, our inequality becomes: 4t^2 - 12t + 5 < 0. See how the “>” flipped to a “<“? That’s a super important detail! Now we've got a slightly friendlier quadratic to deal with. This step was all about making the equation more manageable, like decluttering your workspace before starting a project.

Now, we need to factor this quadratic. Factoring is like finding the hidden ingredients that make up the equation. In this case, we can factor 4t^2 - 12t + 5 into (2t - 1)(2t - 5) < 0. If factoring isn't your strong suit, don't worry, there are plenty of resources to help you brush up. But for now, trust me on this one! Factoring gives us the “roots” of the equation – the values of t that make the expression equal to zero. These roots are super important because they're the boundaries of our time interval. In this case, the roots are t = 1/2 and t = 5/2. Think of these roots as the entry and exit points for our stick's journey above 20 feet. Now, we're getting somewhere!

Determining the Time Interval

We've found the roots of our quadratic, which are t = 1/2 and t = 5/2. What do these numbers mean in the context of our problem? Well, they're the times when the stick is exactly 20 feet in the air – once on the way up, and once on the way down. But we want to know when the stick is above 20 feet, right? This is where a little number line magic comes in handy.

Imagine a number line stretching out, with 1/2 and 5/2 marked on it. These two points divide the number line into three sections: times before 1/2 second, times between 1/2 second and 5/2 seconds, and times after 5/2 seconds. We need to figure out which of these sections represents the time when the stick is higher than 20 feet. One way to do this is to pick a test value from each section and plug it back into our inequality, 4t^2 - 12t + 5 < 0. If the inequality is true for that test value, then that whole section is part of our solution.

Let's try t = 0 (before 1/2 second). Plugging it in, we get 4(0)^2 - 12(0) + 5 < 0, which simplifies to 5 < 0. That's false, so times before 1/2 second aren't part of our interval. Now let's try t = 1 (between 1/2 and 5/2 seconds). We get 4(1)^2 - 12(1) + 5 < 0, which simplifies to -3 < 0. That's true! So, the times between 1/2 second and 5/2 seconds are part of our interval. Finally, let's try t = 3 (after 5/2 seconds). We get 4(3)^2 - 12(3) + 5 < 0, which simplifies to 5 < 0. Again, that’s false. So, times after 5/2 seconds aren't part of our interval either.

This means the stick is above 20 feet between 1/2 second and 5/2 seconds. This is our time interval! We've done it! See how using the number line and test values helped us visualize the solution and make sure we got it right? It’s like using a map to navigate a tricky route – it keeps us from getting lost.

Expressing the Solution

We've figured out that the stick is above 20 feet between t = 1/2 seconds and t = 5/2 seconds. But how do we write that as a mathematical expression? There are a couple of ways, and it's good to know them both.

First, we can use inequality notation. This is a super clear and concise way to say it. We write: 1/2 < t < 5/2. This reads as “t is greater than 1/2 and less than 5/2”. It’s like saying, “The time is trapped between these two numbers.” The < signs tell us we're not including the exact moments when the stick is at 20 feet, just the times when it's above it. This notation is really handy because it directly shows the range of values that satisfy our condition.

Another way to express this is using interval notation. This is a slightly different style, but it means the same thing. We write: (1/2, 5/2). The parentheses mean we're not including the endpoints (1/2 and 5/2) themselves. If we wanted to include the endpoints, we'd use square brackets [ ]. Interval notation is a common shorthand in math, and it's good to get comfortable with it. It's like using a code that other math whizzes will instantly understand.

So, whether you use inequality notation (1/2 < t < 5/2) or interval notation (1/2, 5/2), you're saying the same thing: The stick spends its time above 20 feet between half a second and two and a half seconds after Tim throws it. That’s a pretty good hang time for a stick, right? We’ve taken a real-world problem, translated it into math, solved it, and expressed the answer in a couple of different ways. High five!

Conclusion

Alright, guys, we did it! We successfully found the time interval during which the stick is more than 20 feet above the ground. We started with a word problem, turned it into an inequality, solved the inequality, and then expressed our answer in a couple of different ways. That’s a full math journey right there!

To recap, we learned that the stick is above 20 feet between 1/2 second and 5/2 seconds, or in mathematical terms, 1/2 < t < 5/2 or (1/2, 5/2). But more than just getting the right answer, we practiced some really important problem-solving skills. We learned how to translate a real-world situation into a mathematical model, how to work with inequalities, and how to interpret the results in the context of the original problem. These skills aren't just useful for math class; they're useful for life! Thinking critically, breaking down problems, and finding solutions are things you’ll use every day, no matter what you do.

So, next time you see a stick flying through the air, you can think about the math behind its motion. You can imagine the height function, the parabola it traces, and the time intervals it spends at different altitudes. Maybe you'll even impress your friends with your knowledge of quadratic inequalities! But most importantly, remember that math isn't just about numbers and equations; it's about understanding the world around us. Keep exploring, keep questioning, and keep having fun with it!