In the measurement of the angle of a prism using a spectrometer, the readings of first reflected image are vernier I : `320^(@) 40'` , vernier II : `140^(@) 30'` and those of the second reflected image are vernier I : `80^(@) 38'` , vernier II : `260^(@)` 24'. Then, the angle of the prism is

Options

A. `59^(@)58'`
B. `59^(@)56'`
C. `60^(@)2'`
D. `60^(@)4'`

Explanation

To find the angle of the prism using the given readings from a spectrometer, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Readings:**
- For the first reflected image:
- Vernier I: \(320^\circ 40'\)
- Vernier II: \(140^\circ 30'\)
- For the second reflected image:
- Vernier I: \(80^\circ 38'\)
- Vernier II: \(260^\circ 24'\)

2. **Calculate the Angles (φ1 and φ2):**
- **For φ1:**
\[ φ_1 = x_2 - x_1 \]
Where:
- \(x_2 = 80^\circ 38'\) (from the second reflected image)
- \(x_1 = 320^\circ 40'\) (from the first reflected image)

Performing the subtraction:
\[ φ_1 = 80^\circ 38' - 320^\circ 40' \]
To perform this subtraction, we can convert the angles into minutes:
- \(80^\circ 38' = 80 \times 60 + 38 = 4820\) minutes
- \(320^\circ 40' = 320 \times 60 + 40 = 19240\) minutes

Now, subtract:
\[ φ_1 = 4820 - 19240 = -14420 \text{ minutes} \]
Since we are dealing with angles, we can add \(360^\circ\) (or \(21600\) minutes) to get a positive angle:
\[ φ_1 = -14420 + 21600 = 7180 \text{ minutes} = 119^\circ 58' \]

- **For φ2:**
\[ φ_2 = y_2 - y_1 \]
Where:
- \(y_2 = 260^\circ 24'\) (from the second reflected image)
- \(y_1 = 140^\circ 30'\) (from the first reflected image)

Performing the subtraction:
\[ φ_2 = 260^\circ 24' - 140^\circ 30' \]
Converting to minutes:
- \(260^\circ 24' = 15624\) minutes
- \(140^\circ 30' = 8430\) minutes

Now, subtract:
\[ φ_2 = 15624 - 8430 = 7194 \text{ minutes} \]
Converting back to degrees:
\[ φ_2 = 119^\circ 54' \]

3. **Calculate the Total Angle (φ):**
\[ φ = φ_1 + φ_2 = 119^\circ 58' + 119^\circ 54' \]
Converting to minutes:
\[ φ = (119 \times 60 + 58) + (119 \times 60 + 54) = 7198 + 7174 = 14372 \text{ minutes} \]
Converting back to degrees:
\[ φ = 239^\circ 52' \]

4. **Calculate the Angle of the Prism (A):**
\[ A = \frac{φ}{2} = \frac{239^\circ 52'}{2} \]
Converting to minutes:
\[ A = \frac{14372}{2} = 7186 \text{ minutes} \]
Converting back to degrees:
\[ A = 59^\circ 58' \]

### Final Answer:
The angle of the prism is \(59^\circ 58'\).

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