Stone's Trajectory: Max Height After The Throw

Hey guys! Ever wondered about the math behind throwing a stone off a bridge? Let's dive into a cool problem where Amir throws a stone, and we'll figure out its highest point using some neat equations. We're going to break down the stone's journey step by step – think of it like we're unraveling a real-life physics puzzle!

The Stone's Story: Height Over Time

So, the story begins with Amir throwing a stone, and its height above the water at any given time is described by a quadratic equation. This equation, $h(t)=-5 t^2+20 t+160$, might look a bit intimidating at first, but don't worry! It's just a way of saying that the stone's height (h) depends on how much time (t) has passed since Amir threw it. The equation includes a squared term ($t^2$), which tells us the stone's path is a curve – it goes up and then comes down, thanks to gravity. This curve is called a parabola, and it’s the key to finding the stone's highest point. Understanding this initial equation is crucial, guys, because it's the foundation for everything else we're going to do. We can think of it like the blueprint of the stone's flight. The -5t^2 part represents the effect of gravity pulling the stone down, the 20t shows the initial upward push Amir gives the stone, and the 160? That's the starting height – the height of the bridge above the water. By grasping each piece of the equation, we start to see the whole picture of the stone's journey. Remember, in these kinds of problems, each number and symbol has a meaning. They're telling us the story of the stone’s flight, mathematically! And once we understand the story, we can answer some really interesting questions about it, like when it reaches its peak.

Finding the Peak: Completing the Square

Now, here’s where things get interesting! Amir wants to know when the stone reaches its highest point. In math terms, we need to find the maximum value of the height function, $h(t)$. One super-smart way to do this is by using a technique called "completing the square." It might sound like some fancy math magic, but it's really just a way of rewriting our equation into a form that makes it easy to spot the highest point. Completing the square helps us transform the original equation into vertex form, which gives us the vertex (or turning point) of the parabola directly. This is a powerful tool because the vertex represents either the maximum or minimum point of the quadratic function, and in our case, it's the maximum height of the stone. Here's how we can break it down: First, we factor out the coefficient of the $t^2$ term, which is -5. This gives us a new equation that's a little easier to work with. Then, we manipulate the equation by adding and subtracting a specific value inside the parentheses – this is the "completing the square" part. It might seem a bit tricky, but the goal is to create a perfect square trinomial, which we can then factor into a squared binomial. Once we've done that, the equation is in vertex form, and the coordinates of the vertex are staring right back at us! The x-coordinate of the vertex tells us the time (t) at which the stone reaches its peak height, and the y-coordinate tells us the maximum height itself. This method is not just about finding an answer; it’s about understanding the structure of the quadratic equation and how different forms of the equation reveal different information. So, by mastering completing the square, we can unlock a deeper understanding of these types of problems. And that’s pretty awesome, right?

The Vertex Unveiled: Peak Time and Height

Let's get our hands dirty and actually complete the square for our equation, $h(t)=-5 t^2+20 t+160$. First, we factor out the -5 from the terms involving t: $h(t) = -5(t^2 - 4t) + 160$. Now, we need to complete the square inside the parentheses. We take half of the coefficient of the t term (-4), which is -2, square it to get 4, and then add and subtract it inside the parentheses: $h(t) = -5(t^2 - 4t + 4 - 4) + 160$. Next, we rewrite the perfect square trinomial as a squared binomial and distribute the -5 to the -4 outside the parentheses: $h(t) = -5((t - 2)^2 - 4) + 160 = -5(t - 2)^2 + 20 + 160$. Finally, we simplify the equation to vertex form: $h(t) = -5(t - 2)^2 + 180$. Now, the vertex form of a quadratic equation is $h(t) = a(t - h)^2 + k$, where (h, k) is the vertex. So, in our case, the vertex is (2, 180). This tells us that the stone reaches its maximum height at t = 2 seconds, and the maximum height is 180 meters. Isn’t that cool? We’ve not only found the answer but also understood the entire process of how the stone’s height changes over time. The vertex is a super important point on the parabola, representing the peak of the stone's flight. By transforming our equation into vertex form, we make this point visible, giving us instant access to the maximum height and the time at which it occurs. This method highlights the power of algebraic manipulation in solving real-world problems. We took a complex equation and, through a series of steps, revealed the key information we were looking for. It’s like being a detective, using mathematical tools to uncover the secrets hidden within the equation. And that's why math can be so fascinating – it gives us the ability to model and understand the world around us!

Decoding the Result: What Does It Mean?

Okay, so we've crunched the numbers and found that the stone reaches its peak at 2 seconds, soaring to a height of 180 meters. But what does this really mean in the context of Amir's throw? Let’s decode the result and understand the physical implications. The fact that the peak height is 180 meters tells us that the stone travels quite a distance upwards from the bridge before it starts to fall back down. Remember, the initial height of the stone (when t = 0) was 160 meters, as given in the original equation. So, the stone goes an additional 20 meters above the bridge before gravity takes over. The time it takes to reach this peak, 2 seconds, gives us a sense of the stone's upward velocity. It's a relatively short amount of time, which suggests Amir gave the stone a good, strong throw upwards. These two pieces of information – the peak height and the time to reach it – paint a vivid picture of the stone's trajectory. We can imagine the stone shooting upwards, slowing down as it fights against gravity, reaching its highest point, and then beginning its descent. This whole process is beautifully captured by the quadratic equation, which acts as a mathematical model of a real-world event. But understanding the math is only half the battle. It’s equally important to interpret the results in a meaningful way. This means connecting the numbers back to the physical situation, thinking about what they tell us about the stone's motion, and visualizing the stone's path. By doing this, we transform the problem from a purely mathematical exercise into a story about a stone, a throw, and the power of physics. And that’s what makes problem-solving truly rewarding – when we can see the real-world implications of our mathematical work.

Wrapping Up: The Power of Math in Action

So, guys, we've taken a simple scenario – Amir throwing a stone – and used math to uncover some fascinating details about its journey. We started with a quadratic equation, learned how to complete the square, found the vertex, and then interpreted our results in a real-world context. This is a perfect example of how math can be used to model and understand the world around us. By breaking down the problem into smaller steps, we were able to tackle what seemed like a complex question with confidence. We saw how the equation $h(t)=-5 t^2+20 t+160$ wasn't just a jumble of symbols, but a story waiting to be told. Each term in the equation had a meaning, each step in our solution revealed a new piece of information, and ultimately, we were able to answer Amir's question about the stone's highest point. This process highlights the power of mathematical tools and techniques. Completing the square, in particular, is a versatile method that can be applied to many different types of problems. But beyond the specific techniques, this problem also demonstrates the importance of critical thinking and problem-solving skills. It's not enough to just memorize formulas; we need to understand the underlying concepts and be able to apply them creatively. And that’s what makes math such a valuable skill, whether you’re throwing stones off bridges, designing buildings, or analyzing data. It gives us a framework for thinking logically, solving problems, and making sense of the world. So, next time you see a mathematical equation, don't be intimidated. Think of it as a puzzle waiting to be solved, a story waiting to be told. You might be surprised at what you discover!