- Identify the symmetry in the horizontal distances relative to the vertical distances.
- Calculate the midpoint of the vertical distances that correspond to the same horizontal distance to find the axis of symmetry.
- Recognize that in the parabolic model \(h = a(v - k)^2 + c\), the parameter \(k\) corresponds to the axis of symmetry.
- Conclude that the axis of symmetry at \(v = 3\) gives \(k = 3\), so the correct answer is \(k=3\).
- Final answer: \(\boxed{k=3}\).
Explanation
- Analyzing the Satellite Dish Data
The problem asks us to determine the value of the parameter \(k\) in a parabolic model of a satellite dish based on provided data points relating horizontal distance \(h\) and vertical distance \(v\). The data points given are \((0.41, 0)\), \((0.2, 1.5)\), \((0.07, 3)\), \((0.2, 4.5)\), and \((0.41, 6)\). A satellite dish's parabolic shape implies that its cross-section can be described by a quadratic equation, typically in the form \(h = a(v - k)^2 + c\), where \((k, c)\) represents the vertex of the parabola. We need to find the value of \(k\).
- Identifying the Axis of Symmetry
To find the axis of symmetry, we look for vertical distances \(v\) that share the same horizontal distance \(h\).
- At \(v = 0\) and \(v = 6\), the horizontal distance is \(h = 0.41\).
- At \(v = 1.5\) and \(v = 4.5\), the horizontal distance is \(h = 0.2\).
The axis of symmetry for a parabola is the vertical line passing through the midpoint of any two points on the parabola that share the same horizontal value.
- The midpoint of \(v = 0\) and \(v = 6\) is \(\frac{0 + 6}{2} = 3\).
- The midpoint of \(v = 1.5\) and \(v = 4.5\) is \(\frac{1.5 + 4.5}{2} = 3\).
Consistent results confirm that the axis of symmetry is \(v = 3\).
- Concluding the Value of k
In the equation \(h = a(v - k)^2 + c\), the parameter \(k\) denotes the coordinate of the axis of symmetry on the vertical axis (the horizontal shift of the vertex). Since we established that the axis of symmetry occurs at \(v = 3\), it follows that \(k = 3\). Comparing this result with the provided options, we see that \(k = 3\) is the correct value.
Examples
Satellite dishes are designed with a parabolic shape to focus incoming signals onto a single point called the receiver. Engineers use quadratic equations to model these dishes to ensure they are shaped precisely. By identifying the axis of symmetry and the vertex of the parabola using data points from the physical dish, designers can calculate the optimal placement for the feed horn, which is critical for strong signal reception. Understanding how to model these shapes using the parameter \(k\) helps in manufacturing dishes that perform optimally in real-world conditions.