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 degrees Fahrenheit when the temperature on a new scale is 45 degrees. The ice point and steam point on this new scale are given as 10 degrees and 80 degrees, respectively.

### Step-by-Step Solution:

1. **Identify the relationship between the new scale and the Celsius scale:**
- The ice point (0°C) corresponds to 10° on the new scale.
- The steam point (100°C) corresponds to 80° on the new scale.

2. **Set up the equation for the new scale:**
- Let \( T \) be the temperature in degrees Celsius corresponding to 45° on the new scale.
- We can set up the following proportion based on the ice point and steam point:
\[ \frac{T - 0}{100 - 0} = \frac{45 - 10}{80 - 10} \]

3. **Simplify the equation:**
- The right side simplifies to:
\[ \frac{T}{100} = \frac{35}{70} \]
- This further simplifies to:
\[ \frac{T}{100} = \frac{1}{2} \]

4. **Solve for \( T \):**
- Cross-multiplying gives:
\[ T = \frac{1}{2} \times 100 = 50 \]
- Thus, \( T = 50°C \).

5. **Convert Celsius to Fahrenheit:**
- Use the formula to convert Celsius to Fahrenheit:
\[ F = \frac{9}{5}C + 32 \]
- Substitute \( C = 50 \):
\[ F = \frac{9}{5} \times 50 + 32 \]

6. **Calculate \( F \):**
- First, calculate \( \frac{9}{5} \times 50 \):
\[ \frac{9 \times 50}{5} = 90 \]
- Now add 32:
\[ F = 90 + 32 = 122 \]

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