Designing a Satellite Dish

📌 CONCEPT: The problem of designing a smaller paraboloid that collects only a quarter of the signals can be solved using the concept of similarity in conic sections, where the ratio of corresponding dimensions of two similar figures is the same.

📐 RULE / FORMULA: The formula for similarity in this case is (Depth) ∝ (Width)^(3/2), where the depth remains the same for both the original and the smaller paraboloid, and we need to find the new width. This implies that the ratio of the depths is equal to the ratio of the widths raised to the power of 3/2.

💡 WORKED EXAMPLE: Given the original paraboloid has a depth of 3 meters and a width of 4 meters, we want to find the width of a smaller paraboloid that collects only a quarter of the signals. Let's assume the smaller paraboloid has a depth of 3 meters. We can set up the proportion: 3/3 = 4/x^(3/2), where x is the width of the smaller paraboloid. Solving for x, we get x = (4)^(3/2) / 1 = 8 meters. Therefore, the smaller paraboloid should have a width of 8 meters.

⚠️ COMMON MISTAKE: Students often forget to consider the power of 3/2 in the proportionality formula, which leads to incorrect calculations and ratios.