The focus of a parabolic mirror is at a distance of 6 cm from its vertex. If the mirror is 20 cm deep, find its diameter.

Answer

To solve the problem step by step, we will use the properties of a parabola and the information given in the question.

### Step 1: Understand the Parabola
The equation of a parabola that opens to the right is given by:
\[ y^2 = 4ax \]
where \( a \) is the distance from the vertex to the focus.

### Step 2: Identify Given Values
From the problem:
- The distance from the vertex to the focus (a) = 6 cm
- The depth of the mirror (which corresponds to the x-coordinate) = 20 cm

### Step 3: Substitute Values into the Parabola Equation
Using the values we have:
- \( a = 6 \)
- \( x = 20 \)

Substituting these into the equation:
\[ y^2 = 4 \cdot 6 \cdot 20 \]

### Step 4: Calculate \( y^2 \)
Now calculate \( y^2 \):
\[ y^2 = 4 \cdot 6 \cdot 20 = 480 \]

### Step 5: Find \( y \)
To find \( y \), take the square root of both sides:
\[ y = \sqrt{480} \]

### Step 6: Simplify \( y \)
Calculating \( \sqrt{480} \):
\[ y = \sqrt{16 \cdot 30} = 4\sqrt{30} \]
Using a calculator, \( \sqrt{30} \approx 5.477 \), thus:
\[ y \approx 4 \cdot 5.477 \approx 21.908 \, \text{cm} \]

### Step 7: Calculate the Diameter
The diameter of the mirror is twice the value of \( y \):
\[ \text{Diameter} = 2y = 2 \cdot 21.908 \approx 43.816 \, \text{cm} \]

### Final Answer
The diameter of the parabolic mirror is approximately:
\[ \text{Diameter} \approx 43.82 \, \text{cm} \]

---