Max Height Of RC Plane: Parabolic Flight Calculation

Hey guys! Ever wondered how to calculate the peak altitude of a remote control airplane when it performs a loop in the sky? Imagine an RC plane soaring through the air, tracing a beautiful arc that looks like a parabola. This article will break down a cool math problem involving just that! We'll use a quadratic function to model the plane's flight path and find out its highest point. So, buckle up and let's dive into the fascinating world of parabolic flights and mathematical calculations!

Understanding the Problem

In this scenario, we're looking at a remote-controlled airplane performing an aerial stunt. The plane's trajectory follows a parabolic path, which is described by the quadratic function y = -x² + 10x. Here, 'y' represents the altitude of the plane, and 'x' represents its horizontal position. Our main goal is to find the maximum height the plane reaches during this maneuver. This problem beautifully blends the practical application of physics with mathematical concepts, making it super engaging and relevant. You might be thinking, why is this useful? Well, understanding parabolic motion is crucial in many fields, from engineering and sports to even video game design! Think about how a basketball travels through the air when you shoot a hoop, or the path of a projectile launched from a cannon – they all follow parabolic trajectories. So, grasping this concept can open doors to many cool applications.

To solve this, we need to tap into our knowledge of quadratic functions and parabolas. Remember, a parabola is a U-shaped curve, and in our case, since the coefficient of the x² term is negative (-1), the parabola opens downwards. This means it has a maximum point, which is exactly what we're trying to find – the highest altitude the plane reaches. Now, how do we find this maximum point? This is where understanding the properties of a parabola becomes key. The maximum or minimum point of a parabola is called its vertex. For a quadratic function in the form y = ax² + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. Once we have the x-coordinate, we can plug it back into the original equation to find the y-coordinate, which represents the maximum height in our case. So, let's put on our math hats and get ready to calculate!

Breaking Down the Quadratic Function

Let's dive deeper into the quadratic function y = -x² + 10x. This equation is the heart of our problem, and understanding its components is crucial to finding the solution. First off, a quadratic function is a polynomial function of degree two, meaning the highest power of the variable (in this case, 'x') is 2. The general form of a quadratic function is y = ax² + bx + c, where 'a', 'b', and 'c' are constants. In our specific equation, we can identify these constants: a = -1, b = 10, and c = 0 (since there's no constant term explicitly written). The coefficient 'a' plays a significant role in determining the shape of the parabola. As we mentioned earlier, if 'a' is negative, the parabola opens downwards, indicating a maximum point. If 'a' is positive, the parabola opens upwards, indicating a minimum point. In our case, a = -1, confirming that we're dealing with a parabola that opens downwards and has a maximum height. The coefficient 'b' influences the position of the parabola's axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. The vertex, which represents the maximum or minimum point, always lies on this axis of symmetry. Understanding the relationship between these coefficients and the shape and position of the parabola is fundamental to solving problems like this. It allows us to visualize the plane's flight path and anticipate where the maximum height will occur. So, with this foundation in place, we're well-equipped to move on to the next step: finding the vertex.

Finding the Vertex: The Key to Maximum Height

The vertex is the holy grail here, guys! It's the point where the parabola changes direction, and in our case, it represents the maximum altitude the RC plane reaches. To pinpoint the vertex, we'll use a nifty formula that leverages the coefficients of our quadratic equation. Remember the general form, y = ax² + bx + c? The x-coordinate of the vertex can be calculated using the formula: x = -b / 2a. This formula is a direct result of completing the square or using calculus to find the critical point of the quadratic function. It's a powerful tool that simplifies the process of finding the vertex. Now, let's plug in the values from our equation, y = -x² + 10x. We identified earlier that a = -1 and b = 10. Substituting these values into the formula, we get: x = -10 / (2 * -1) x = -10 / -2 x = 5 So, the x-coordinate of the vertex is 5. But what does this mean in the context of our problem? Well, it tells us the horizontal position at which the plane reaches its maximum height. It's like saying, the plane reaches its peak altitude 5 units along the x-axis. However, we're ultimately interested in the maximum height itself, which is the y-coordinate of the vertex. To find the y-coordinate, we simply substitute the x-value we just calculated (x = 5) back into the original equation. This will give us the altitude (y) at that particular horizontal position. So, let's do that in the next step!

Calculating the Y-Coordinate

Alright, we've found the x-coordinate of the vertex, which is 5. Now, let's find the corresponding y-coordinate, which will give us the maximum height the plane reaches. To do this, we'll substitute x = 5 back into our original equation: y = -x² + 10x Plugging in x = 5, we get: y = -(5)² + 10(5) y = -25 + 50 y = 25 So, the y-coordinate of the vertex is 25. This is the magic number we've been looking for! In the context of our problem, this means that the maximum height reached by the remote control airplane is 25 units. Now, let's think about the units. The problem doesn't explicitly state the units, but we can assume they are meters (m), as this is a common unit for measuring altitude in such scenarios. Therefore, the maximum height reached by the plane is 25 meters. Isn't it amazing how we can use a simple quadratic equation to model the flight path of an airplane and calculate its maximum height? This problem showcases the power of mathematics in describing and predicting real-world phenomena. We've successfully navigated through the problem, found the vertex, and determined the maximum height. Now, let's wrap it all up with the final answer and some concluding thoughts.

The Grand Finale: Determining the Maximum Height

Drumroll, please! After all our calculations, we've arrived at the final answer. We found that the y-coordinate of the vertex is 25, which represents the maximum height reached by the RC plane. Assuming the units are meters, the maximum height is 25 meters. Therefore, the correct answer is e. 25 m. Woohoo! We nailed it! This problem might seem a bit complex at first glance, but by breaking it down into smaller, manageable steps, we were able to solve it with ease. We used our understanding of quadratic functions, parabolas, and the vertex formula to find the solution. This is a perfect example of how mathematical concepts can be applied to real-world scenarios. Imagine using this knowledge to design better aerial stunts for RC planes, calculate the trajectory of a projectile, or even optimize the path of a roller coaster! The possibilities are endless. Solving problems like this not only strengthens our mathematical skills but also enhances our problem-solving abilities in general. It teaches us to think critically, analyze information, and apply the right tools to find solutions. So, the next time you see an RC plane soaring through the sky, remember the parabola and the math behind its graceful flight. And who knows, maybe you'll be inspired to explore the fascinating world of mathematics and its applications even further!