Hey guys! Ever wondered how engineers design those awesome arch-shaped structures? Today, we're diving into a cool problem where an engineer is designing a parabolic arch gate for an amusement park. Let's break down the challenge and see how we can find the equation for this arch. It's like a real-world math puzzle, and we're here to solve it together!
The Arch Gate Challenge: Understanding the Problem
So, our engineer has a vision: a grand arch-shaped gate welcoming visitors to the amusement park. This gate needs to be 80 feet wide and 25 feet tall. That's quite the entrance! The key here is that the gate has a parabolic shape. Now, parabolas might sound intimidating, but they're just U-shaped curves that follow a specific mathematical equation. Our mission is to figure out exactly which equation describes the shape of this gate. This involves translating the given dimensions into mathematical coordinates and then using those coordinates to determine the equation of the parabola. We will focus on understanding the properties of parabolas and how they relate to real-world applications. The width and height constraints are crucial pieces of information that will guide us in determining the parameters of the parabolic equation. To kick things off, let's visualize the gate in a coordinate system. This will help us translate the word problem into a visual representation, making it easier to identify the key points and dimensions we need for our equation.
Setting Up the Coordinate System
To make things easier, let's imagine this gate on a coordinate plane. Where should we put the origin (0,0)? A smart move is to place the vertex (the highest point) of the parabola on the y-axis. This simplifies our equation quite a bit. We can also center the arch so that the vertex is at the point (0, 25), as this represents the maximum height of the gate. Since the gate is 80 feet wide, we can place the endpoints of the base at (-40, 0) and (40, 0) on the x-axis. These points represent where the arch meets the ground. By strategically placing the parabola on the coordinate plane, we can leverage the symmetry of the parabola and simplify the calculations involved in finding its equation. Now, with our coordinate system in place, we have three key points: the vertex (0, 25) and the two endpoints (-40, 0) and (40, 0). These points are our anchors, providing the necessary information to determine the specific equation of the parabolic arch. It's like having the corner pieces of a puzzle; now we just need to fill in the rest!
The Parabola Equation: Vertex Form
Alright, let's talk equations! The most helpful form for a parabola when we know the vertex is the vertex form: y = a(x - h)^2 + k. Here, (h, k) is the vertex of the parabola, and 'a' determines how wide or narrow the parabola is, and whether it opens upwards or downwards. Since our vertex is at (0, 25), we can plug those values into the equation: y = a(x - 0)^2 + 25, which simplifies to y = ax^2 + 25. We're getting closer! Now, we just need to find the value of 'a'. Remember, 'a' is the key to defining the shape of our parabola. A larger absolute value of 'a' means a narrower parabola, while a smaller value means a wider one. The sign of 'a' tells us whether the parabola opens upwards (positive 'a') or downwards (negative 'a'). In our case, since the arch opens downwards, we know 'a' will be negative. To find the exact value of 'a', we'll use one of the points on the parabola that we identified earlier. This will allow us to create an equation with 'a' as the only unknown, which we can then solve.
Finding the Value of 'a'
To find 'a', we can use one of the endpoints, say (40, 0). We substitute these coordinates into our equation y = ax^2 + 25: 0 = a(40)^2 + 25. Now we have a simple equation to solve for 'a'! Let's break it down: 0 = 1600a + 25. Subtracting 25 from both sides gives us -25 = 1600a. Now, divide both sides by 1600: a = -25/1600. We can simplify this fraction by dividing both numerator and denominator by 25: a = -1/64. So, we've found our 'a'! This value tells us how stretched or compressed our parabola is. A negative 'a' confirms that the parabola opens downwards, as we expected. Now that we have 'a', we have all the pieces we need to write the final equation of our parabolic arch. It's like finding the last piece of a jigsaw puzzle – everything clicks into place!
The Final Equation
We've done it! We found 'a' which is -1/64. Now, we plug this back into our equation y = ax^2 + 25 to get the final equation for the parabolic arch: y = (-1/64)x^2 + 25. But wait! The options given are in a slightly different form. They look like x^2 = .... So, let's rearrange our equation to match. First, subtract 25 from both sides: y - 25 = (-1/64)x^2. Next, multiply both sides by -64: -64(y - 25) = x^2. Finally, we can rewrite this as: x^2 = -64(y - 25). And there we have it! This is the equation of the parabolic arch gate in the desired format. This equation precisely describes the shape of the arch, ensuring it meets the engineer's specifications for width and height. It's a testament to the power of mathematics in real-world design and engineering.
Choosing the Correct Option
Looking at the options, the correct equation is the one that matches our derived equation: x^2 = -64(y - 25). This corresponds to option A. Now, it's important to understand why this equation works. The negative sign in front of the 64 indicates that the parabola opens downwards. The (y - 25) term shifts the vertex of the parabola upwards to the point (0, 25), which is the specified height of the gate. The coefficient 64 determines the width of the parabola; a larger coefficient would result in a narrower parabola, while a smaller coefficient would result in a wider one. By carefully considering each part of the equation, we can see how it accurately represents the shape and dimensions of the arch gate.
Real-World Applications of Parabolas
Isn't it amazing how math connects to the real world? Parabolas aren't just abstract shapes; they're everywhere! Think about satellite dishes, which use the parabolic shape to focus signals. Car headlights use parabolic reflectors to direct light. Even the path of a ball thrown through the air roughly follows a parabolic curve (ignoring air resistance, of course!). Understanding parabolas is super useful in engineering, physics, and even architecture. They provide optimal shapes for structures that need to distribute weight evenly, like bridges and arches. The principles we've discussed in this problem can be applied to a wide range of design challenges, from small-scale projects to large-scale infrastructure. By mastering the properties of parabolas, engineers and designers can create efficient, aesthetically pleasing, and structurally sound solutions.
Conclusion: Math in Action
So, we successfully navigated the challenge of finding the equation for the parabolic arch gate! We started by understanding the problem, visualizing it on a coordinate plane, using the vertex form of a parabola equation, solving for the unknown parameter 'a', and finally, rearranging the equation to match the given options. This problem shows how math is a powerful tool for solving real-world design challenges. By applying mathematical principles, engineers can create structures that are both functional and beautiful. And remember, the next time you see an arch or a satellite dish, you'll know there's some cool math behind its shape! Keep exploring the world around you, and you'll find math in action everywhere you look.
A. x^2=-64(y-25)
