The focal length of a convex mirror is obtained by using a convex lens The following observations are recorded during theexperiment .
`{:("objectopsition",,=5cm),("lens",,=35.4cm),("Image",,=93.8cm),("Mirror",,=63.3cm),("Bencherror",,=-0.1cm):}`
then the focal length of mirror will be .

Options

A. `7.5`
B. `8.4cm`
C. `15.3cm`
D. None of these

Answer

To find the focal length of a convex mirror using the provided observations, we can follow these steps:

### Step 1: Understand the Given Data
We have the following observations:
- Object position (u) = -5 cm (the object distance is taken as negative in lens/mirror formula)
- Lens position (v_lens) = 35.4 cm
- Image position (v_image) = 93.8 cm
- Mirror position (v_mirror) = 63.3 cm
- Bench error = -0.1 cm

### Step 2: Calculate the Distance from the Image to the Mirror
To find the distance from the image to the mirror, we need to calculate:
\[ R = v_{image} - v_{mirror} \]
Substituting the values:
\[ R = 93.8 \, \text{cm} - 63.3 \, \text{cm} = 30.5 \, \text{cm} \]

### Step 3: Account for the Bench Error
Since there is a bench error of -0.1 cm, we need to adjust the calculated radius of curvature:
\[ R_{true} = R + \text{bench error} \]
\[ R_{true} = 30.5 \, \text{cm} - 0.1 \, \text{cm} = 30.4 \, \text{cm} \]

### Step 4: Calculate the Focal Length
The focal length (f) of a convex mirror is related to the radius of curvature (R) by the formula:
\[ f = \frac{R}{2} \]
Substituting the true radius of curvature:
\[ f = \frac{30.4 \, \text{cm}}{2} = 15.2 \, \text{cm} \]

### Final Answer
Thus, the focal length of the convex mirror is:
\[ f = 15.2 \, \text{cm} \]

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