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 use the mirror formula and the magnification formula.

### Step 1: Calculate the Focal Length
The focal length (f) of a concave mirror is given by the formula:
\[ f = \frac{R}{2} \]
where \( R \) is the radius of curvature.

Given:
\[ R = 35.0 \, \text{cm} \]

Calculating the focal length:
\[ f = \frac{35.0 \, \text{cm}}{2} = 17.5 \, \text{cm} \]

### Step 2: Use the Magnification Formula
The magnification (m) is defined as:
\[ m = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance.

Given that the image is 2.5 times the size of the object (upright image), we have:
\[ m = 2.5 \]

### Step 3: Relate Image Distance to Object Distance
From the magnification formula, we can express \( v \) in terms of \( u \):
\[ v = -m \cdot u \]
Substituting the value of \( m \):
\[ v = -2.5u \]

### Step 4: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Substituting \( v = -2.5u \) into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-2.5u} + \frac{1}{u} \]

### Step 5: Simplify the Equation
Rearranging the equation:
\[ \frac{1}{f} = \frac{1 - (-2.5)}{2.5u} = \frac{1 + 2.5}{2.5u} = \frac{3.5}{2.5u} \]

Now, substituting the value of \( f \):
\[ \frac{1}{-17.5} = \frac{3.5}{2.5u} \]

### Step 6: Solve for \( u \)
Cross-multiplying gives:
\[ -17.5 \cdot 3.5 = 2.5u \]
Calculating the left side:
\[ -61.25 = 2.5u \]

Now, solving for \( u \):
\[ u = \frac{-61.25}{2.5} = -24.5 \, \text{cm} \]

### Step 7: Calculate the Distance from the Mirror
Since the object distance \( u \) is negative (by convention, as the object is in front of the mirror), the distance from the mirror to the face is:
\[ |u| = 24.5 \, \text{cm} \]

Thus, the distance from the mirror to the face is approximately **24.5 cm**.

### Final Answer
The mirror is approximately **24.5 cm** from the face.

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