A parabola reflector is 9 cm deep and its diameter is 24 cm. Find the distance of its focus from vertex.

Answer

To find the distance of the focus from the vertex of a parabola reflector that is 9 cm deep and has a diameter of 24 cm, we can follow these steps:

### Step 1: Understand the Geometry of the Parabola
We can model the parabola using the standard equation \( y^2 = 4ax \). Here, \( a \) represents the distance from the vertex to the focus.

### Step 2: Identify the Dimensions
The depth of the parabola is given as 9 cm, and the diameter is 24 cm. This means the radius (which is half of the diameter) is \( r = \frac{24}{2} = 12 \) cm.

### Step 3: Set Up the Coordinates
Let's place the vertex of the parabola at the origin (0, 0). The parabola opens to the right. The point at the bottom of the parabola (the vertex) is at (0, 0), and the point at the edge of the parabola at the depth of 9 cm is at (9, 12) because the depth is along the x-axis and the radius is along the y-axis.

### Step 4: Substitute the Coordinates into the Parabola Equation
We have the point (9, 12) that lies on the parabola. We can substitute \( x = 9 \) and \( y = 12 \) into the equation \( y^2 = 4ax \):

\[ 12^2 = 4a \cdot 9 \]

### Step 5: Solve for \( a \)
Calculating \( 12^2 \):

\[ 144 = 36a \]

Now, divide both sides by 36:

\[ a = \frac{144}{36} = 4 \]

### Step 6: Conclusion
The distance of the focus from the vertex is equal to \( a \), which we found to be 4 cm.

### Final Answer
The distance of the focus from the vertex is **4 cm**.

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