A long-sighted person cannot see objects clearly at a distance less than 40 cm from his eye. The power of the lens needed to read an object at 25 cm is

Options

A. `-2 .5 D`
B. `+ 2.5 D`
C. `-6.25`D
D. `+ 1.5` D

Answer

To find the power of the lens needed for a long-sighted person to read an object at 25 cm, we can follow these steps:

### Step 1: Identify the Given Values
- The object distance (u) is given as -25 cm (the negative sign indicates that the object is on the same side as the incoming light).
- The image distance (v) is given as -40 cm (this indicates that the image is formed on the same side as the object, which is typical for a long-sighted person).

### Step 2: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step 3: Substitute the Values into the Lens Formula
Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{-40} - \frac{1}{-25} \]

### Step 4: Calculate the Right Side
Calculating the right side:
\[ \frac{1}{f} = -\frac{1}{40} + \frac{1}{25} \]
To combine these fractions, we need a common denominator. The least common multiple of 40 and 25 is 200.
\[ \frac{1}{f} = -\frac{5}{200} + \frac{8}{200} = \frac{3}{200} \]

### Step 5: Find the Focal Length
Now, we can find \( f \):
\[ f = \frac{200}{3} \text{ cm} \approx 66.67 \text{ cm} \]

### Step 6: Calculate the Power of the Lens
The power \( P \) of a lens is given by the formula:
\[ P = \frac{1}{f} \text{ (in meters)} \]
First, convert \( f \) from cm to meters:
\[ f = \frac{200}{3} \text{ cm} = \frac{200}{300} \text{ m} = \frac{2}{3} \text{ m} \]
Now, substituting into the power formula:
\[ P = \frac{1}{\frac{2}{3}} = \frac{3}{2} = 1.5 \text{ diopters} \]

### Conclusion
Therefore, the power of the lens needed for the long-sighted person to read an object at 25 cm is +1.5 diopters.

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