#### 30Question: A satellite dish is shaped like a parabola with a focus at (0, 5). If the dish's opening is 20 units wide, find the equation of the parabola in standard form.
["Solving the Satellite Dish Parabola: Finding the Equation Using Key Details", "In the design of satellite dishes, geometry plays a crucial role in ensuring optimal signal focus. One fundamental shape used is the parabola, prized for its ability to reflect incoming parallel rays to a single point — the focus. Understanding how to derive the equation of such a parabola is essential for engineers and students alike.", "---", "### What You Need to Know", "A satellite dish is modeled as a parabola, where:", "- The focus is a point that all reflected signals converge to.\n- The directrix is a line defining the reflective boundary.\n- The vertex is halfway between the focus and directrix for standard satellite dishes.", "In this article, we explore how to determine the standard-form equation of a parabolic satellite dish given key parameters: a vertical focus at (0, 5) and a width of 20 units across the opening.", "---", "### Step 1: Place the Parabola on the Coordinate Plane", "Since the focus is at (0, 5) and the dish opening is symmetric about the y-axis, we model the parabola as opening upward or downward.", "Use the vertex form of a vertical parabola:\n[\ny = a(x - h)^2 + k\n]", "The vertex ((h, k)) lies midway between the focus and the directrix. Because the focus is at (0, 5), and assuming the dish opens symmetrically around the y-axis, the vertex is directly below the focus along the axis of symmetry.", "Let’s denote the vertex as ((0, k)). The distance from vertex to focus is ( |k - 5| = p ), called the focal length.", "---", "### Step 2: Use Given Focus and Width to Find Vertex", "The dish opening is 20 units wide, meaning the diameter of the parabola at a certain height is 20. Because the parabola is symmetric, the two endpoints of the opening lie at ( y = 5 + p \pm 10 ), but since the focus is above vertex, and the dish opens upward, the lowest point (vertex) must be halfway — so if the focus is at (0, 5), and the width spans 20 units symmetrically around the vertex, then:", "Let the vertex be at ((0, k)). Then the two endpoints of the opening are at ((-10, k + 10)) and ((10, k + 10)), since width = 20.", "But these points must lie on the parabola. Using the standard form ( y = a x^2 + k ) (vertex at origin shifted up), and knowing the vertex is at ((0, k)), the general form centered at vertex is:", "[\ny = a x^2 + k\n]", "Insert one endpoint, say ((10, k + 10)):", "[\nk + 10 = a(10)^2 + k \Rightarrow 10 = 100a \Rightarrow a = \frac{1}{10}\n]", "Now compute the focal length ( p = \frac{1}{4a} ):", "[\np = \frac{1}{4 \cdot (1/10)} = \frac{10}{4} = 2.5\n]", "Since the focus is located ( p = 2.5 ) units above the vertex, and the focus is at (0, 5), the vertex must be at:", "[\nk = 5 - 2.5 = 2.5\n]", "So the vertex is at ( (0, 2.5) )", "---", "### Step 3: Write the Equation in Standard Form", "Using vertex form:\n[\ny = \frac{1}{10}x^2 + 2.5\n]\nor rearranged:\n[\ny = \frac{1}{10}x^2 + \frac{5}{2}\n]", "---", "### Final Answer: The Standard-Equation of the Satellite Dish Parabola", "[\n\boxed{y = \frac{1}{10}x^2 + \frac{5}{2}}\n]", "This parabola opens upward, focus at (0, 5), and vertex at (0, 2.5), perfectly modeling the satellite dish’s focusing geometry.", "---", "### Why This Matters", "- Parabolic reflectors concentrate incoming radio or microwave signals to the focus, boosting reception.\n- Knowing the equation helps engineers design precise satellite dishes.\n- This derivation demonstrates how geometry and algebra combine in real-world engineering applications.", "Understanding how to convert physical properties—like focus location and width—into a mathematical equation is key to mastering parabolic design. Whether you're a physics student or an engineering professional, mastering this ensures optimal performance in satellite technology.", "---", "Keywords: satellite dish parabola, standard form parabola equation, vertex form, focal length, satellite reflector geometry, coordinate geometry, focus and directrix, parabolic dish design, math modeling satellites.", "---", "For more on parabolas in engineering, explore our guides on optics, satellite systems, and reflective surfaces."]
