Find the diameter of the image of the moon formed by a spherical concave mirror of focal length 7.6 m. The diameter of the moon is 3450 km and the distance of the earth and the moon is `3.8 xx 10^5 km.`

Explanation

To find the diameter of the image of the moon formed by a spherical concave mirror, we will follow these steps:

### Step 1: Understand the Given Data
- Focal length of the concave mirror, \( F = 7.6 \, \text{m} \)
- Diameter of the moon, \( D = 3450 \, \text{km} \)
- Distance from the Earth to the Moon, \( U = 3.8 \times 10^5 \, \text{km} \)

### Step 2: Convert Units
We need to convert the diameter of the moon and the distance from the Earth to the Moon into meters for consistency in units.
- Convert the diameter of the moon:
\[ D = 3450 \, \text{km} = 3450 \times 10^3 \, \text{m} = 3.45 \times 10^6 \, \text{m} \]
- Convert the distance from the Earth to the Moon:
\[ U = 3.8 \times 10^5 \, \text{km} = 3.8 \times 10^5 \times 10^3 \, \text{m} = 3.8 \times 10^8 \, \text{m} \]

### Step 3: Determine the Image Distance
Since the moon is at a very far distance, we can consider the object distance \( U \) to be effectively infinite. For rays coming from infinity, the image is formed at the focal point of the mirror:
\[ V = F = 7.6 \, \text{m} \]

### Step 4: Use the Magnification Formula
The magnification \( m \) of the mirror is given by:
\[ m = \frac{h_i}{h_o} = -\frac{V}{U} \]
Where:
- \( h_i \) = height of the image (which is the diameter of the image we need to find)
- \( h_o \) = height of the object (diameter of the moon)

### Step 5: Substitute Known Values
Substituting the known values into the magnification formula:
\[ m = -\frac{7.6}{3.8 \times 10^8} \]
Now, we can calculate \( m \):
\[ m = -\frac{7.6}{3.8 \times 10^8} = -2 \times 10^{-8} \]

### Step 6: Calculate the Height of the Image
Now we can find the height of the image using the magnification:
\[ \frac{h_i}{h_o} = -\frac{7.6}{3.8 \times 10^8} \]
Substituting \( h_o = 3.45 \times 10^6 \, \text{m} \):
\[ h_i = h_o \cdot m = 3.45 \times 10^6 \cdot (-2 \times 10^{-8}) \]
Calculating \( h_i \):
\[ h_i = 3.45 \times 10^6 \cdot -2 \times 10^{-8} = -0.069 \, \text{m} \]
Taking the absolute value, since we are interested in the diameter:
\[ h_i = 0.069 \, \text{m} = 6.9 \, \text{cm} \]

### Final Answer
The diameter of the image of the moon formed by the concave mirror is:
\[ \text{Diameter of the image} = 6.9 \, \text{cm} \]