III. DIRECTION: Read, analyze, and solve the following problems:

1. An engineer designs satellite dish with a parolic cross section. The dish is 72 inches wide and 54 inches deep. How far is the receiver from the bottom of the dish if the receiver is placed at the focus?

We are asked to find how far the receiver is from the bottom of a satellite dish, which has a parabolic cross section. The problem describes the dish as being **72 inches wide** and **54 inches deep**, and the receiver is placed at the **focus** of the parabola.

### Key Points:
- The parabola is symmetric, so its vertex is at the lowest point.
- The width of the dish gives the distance between two points on the parabola, 72 inches across. Since the parabola is symmetric, the vertex is at \(x = 0\), and the width implies that at \(x = \pm 36\) inches (half of 72), the dish is 54 inches deep.

The general equation for a parabola with its vertex at the origin and opening upwards is:
\[
y = \frac{x^2}{4p}
\]
where \(p\) is the distance from the vertex to the focus, and \(x\) is the horizontal distance from the vertex.

### Step 1: Use the given point to find \(p\)

At \(x = 36\) inches (half the width), the depth \(y = 54\) inches, so we substitute these into the equation:
\[
54 = \frac{36^2}{4p}
\]
\[
54 = \frac{1296}{4p}
\]
\[
54 = \frac{1296}{4p} \quad \Rightarrow \quad 4p = \frac{1296}{54} = 24
\]
\[
p = \frac{24}{4} = 6
\]

### Step 2: Interpretation

The value of \(p = 6\) inches. This means the **receiver** (focus) is placed **6 inches above the vertex**, which is at the bottom of the dish.

### Final Answer:
The receiver is **6 inches** from the bottom of the dish.

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Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this:

1. How would the distance to the focus change if the dish were wider but not deeper?
2. What is the equation of the parabola if the dish is 72 inches wide and 54 inches deep?
3. How would the focal distance change if the depth of the dish were increased by 10 inches?
4. What is the significance of the focus in a parabolic dish in terms of signal reflection?
5. How can we calculate the area of the parabolic section of the dish?

**Tip:** The distance from the vertex to the focus determines how concentrated signals will be reflected into the receiver!

An engineer designs a satellite dish with a parabolic cross section. The dish is 72 inches wide and 54 inches deep. How far is the receiver from the bottom of the dish if the receiver is placed at the focus?

Discover how to find the distance from the vertex to the focus of a parabolic satellite dish with a width of 72 inches and a depth of 54 inches. This solution explores the geometry of parabolas and uses key equations to determine the placement of the receiver. Ideal for students in Grades 9-11.

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