A concave shaving mirror has a radius of curvature of `35.0 cm`. It is positioned so that the (upright) image of man's face is `2.50` times the size of the face. How far is the mirror from the face ?

Options

A. `5.25 cm`
B. `21.0 cm`
C. `10.5 cm`
D. `42 cm`

Explanation

To solve the problem step by step, we will follow the principles of optics, particularly the mirror formula and magnification.

### Step 1: Identify Given Data
- Radius of curvature (R) = 35.0 cm
- Magnification (M) = 2.50 (the image is upright and enlarged)

### Step 2: Calculate the Focal Length (F)
The focal length (F) of a concave mirror is given by the formula:
\[ F = \frac{R}{2} \]
Substituting the value of R:
\[ F = \frac{35.0 \, \text{cm}}{2} = 17.5 \, \text{cm} \]

### Step 3: Relate Magnification to Object and Image Distances
The magnification (M) is given by the formula:
\[ M = -\frac{V}{U} \]
Where:
- V = Image distance
- U = Object distance

Since the image is upright and enlarged, we can write:
\[ M = \frac{H_i}{H_o} = 2.50 \]
This leads to:
\[ 2.50 = -\frac{V}{U} \implies V = -2.50U \]

### Step 4: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Substituting the value of V from the previous step:
\[ \frac{1}{17.5} = \frac{1}{-2.50U} + \frac{1}{U} \]

### Step 5: Simplify the Equation
To combine the terms on the right side:
\[ \frac{1}{17.5} = \frac{-1 + 2.50}{2.50U} \]
This simplifies to:
\[ \frac{1}{17.5} = \frac{1.50}{2.50U} \]

### Step 6: Cross Multiply to Solve for U
Cross multiplying gives:
\[ 1.50 \cdot 17.5 = 2.50U \]
Calculating the left side:
\[ 26.25 = 2.50U \]
Now, solving for U:
\[ U = \frac{26.25}{2.50} = 10.5 \, \text{cm} \]

### Conclusion
The distance of the mirror from the face (U) is **10.5 cm**.

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