A large temple has a depression in one wall. On the floor plan it appears as a indentation having spherical shape of radius `2.50 m` A worshiper stands on the center line of the depression, `2.00` m out from its deepest point,and whispers a prayers. Where is the sound concentrated after reflection from the back wall of the depression?

Options

A. `(-10)/3m`
B. `(-10)/7m`
C. `(-3)/10m`
D. `(-7)/10m`

Answer

To solve the problem of where the sound is concentrated after reflection from the back wall of the spherical depression, we can follow these steps:

### Step 1: Understand the Geometry
The depression in the wall is spherical with a radius of \( R = 2.50 \, \text{m} \). The worshiper is standing at a distance of \( 2.00 \, \text{m} \) from the deepest point of the depression.

### Step 2: Determine the Object Distance
Since the deepest point of the depression can be considered as the origin, the distance of the worshiper from this point is:
\[ u = -2.00 \, \text{m} \]
(Note: The negative sign indicates that the object is in front of the mirror.)

### Step 3: Find the Focal Length
The focal length \( f \) of a spherical mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the radius:
\[ f = \frac{2.50 \, \text{m}}{2} = 1.25 \, \text{m} \]

### Step 4: Use the Mirror Formula
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{1.25} = \frac{1}{v} + \frac{1}{-2.00} \]

### Step 5: Solve for Image Distance \( v \)
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{1.25} + \frac{1}{2.00} \]
Calculating the left side:
\[ \frac{1}{1.25} = 0.8 \]
\[ \frac{1}{2.00} = 0.5 \]
So,
\[ \frac{1}{v} = 0.8 - 0.5 = 0.3 \]
Thus,
\[ v = \frac{1}{0.3} = \frac{10}{3} \, \text{m} \]

### Step 6: Interpret the Result
The positive value of \( v \) indicates that the sound is concentrated at a distance of \( \frac{10}{3} \, \text{m} \) from the deepest point of the depression, on the same side as the worshiper.

### Final Answer
The sound is concentrated at a distance of \( \frac{10}{3} \, \text{m} \) from the deepest point of the depression.

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