Ice point and steam point on a particular scale reads `10^@ and 80^@` respectively . The temperature on `.^@F` Scale when temperature on new scale is `45^@` is

Options

A. `50^@F`
B. `112^@F`
C. `122^@F`
D. `138^@F`

Answer

To solve the problem, we need to find the temperature in Fahrenheit when the temperature on the new scale is 45°C. The new scale has an ice point of 10°C and a steam point of 80°C. We will follow these steps:

### Step-by-Step Solution:

1. **Identify the given information:**
- Ice point on the new scale = 10°C
- Steam point on the new scale = 80°C
- Temperature on the new scale (Tn) = 45°C

2. **Calculate the range of the new scale:**
- Range = Steam point - Ice point = 80°C - 10°C = 70°C

3. **Determine the temperature per unit division for the new scale:**
- The temperature change per unit division on the new scale is:
\[ \text{Temperature per division} = \frac{\text{Range}}{\text{Number of divisions}} = \frac{70°C}{70} = 1°C \]

4. **Set up the relationship between the new scale and the Celsius scale:**
- Let T be the temperature in Celsius. The relationship can be expressed as:
\[ \frac{Tn - 10}{80 - 10} = \frac{T - 0}{100 - 0} \]
- Substituting the known values:
\[ \frac{45 - 10}{70} = \frac{T - 0}{100} \]
- Simplifying the left side:
\[ \frac{35}{70} = \frac{T}{100} \]
- This simplifies to:
\[ \frac{1}{2} = \frac{T}{100} \]

5. **Solve for T (temperature in Celsius):**
- Cross-multiplying gives:
\[ T = 100 \times \frac{1}{2} = 50°C \]

6. **Convert Celsius to Fahrenheit:**
- Use the formula to convert Celsius to Fahrenheit:
\[ T_F = \frac{9}{5}T + 32 \]
- Substituting T = 50°C:
\[ T_F = \frac{9}{5} \times 50 + 32 \]
- Calculating:
\[ T_F = 90 + 32 = 122°F \]

### Final Answer:
The temperature on the Fahrenheit scale when the temperature on the new scale is 45°C is **122°F**.