Three thermometer are placed in a closed, insulated box and are allowed to reach thermal equilibrium . One is calibrated in Fahrenheit degrees, one in Celsius degrees, and one in Kelvins. The Celsius thermometer reads `-40^@C` and the Kelvin thermometer reads 233 K. Which one of the following statements is necessarily true ?

Options

A. The Kelvin thermometer should read - 233 K
B. The Kelvin thermometer should read -313 K
C. The Fahrenheit thermometer must read `-40^@0` F.
D. If water were found within the box, it must be in the liquid state .

Answer

To solve the problem, we need to analyze the readings from the thermometers and understand the relationships between the different temperature scales.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We have three thermometers: one in Celsius (°C), one in Kelvin (K), and one in Fahrenheit (°F).
- The Celsius thermometer reads -40°C.
- The Kelvin thermometer reads 233 K.

2. **Convert the Kelvin Reading to Celsius:**
- The relationship between Kelvin and Celsius is given by the formula:
\[ T(K) = T(°C) + 273.15 \]
- Rearranging this gives:
\[ T(°C) = T(K) - 273.15 \]
- Substituting the Kelvin reading:
\[ T(°C) = 233 K - 273.15 = -40.15 °C \]
- However, we already know from the problem that the Celsius thermometer reads -40°C, which suggests that the Kelvin thermometer reading is not consistent with the Celsius thermometer reading.

3. **Analyzing the Celsius and Kelvin Readings:**
- Since the Celsius thermometer reads -40°C, we can conclude that the Kelvin thermometer should read:
\[ T(K) = -40 + 273.15 = 233.15 K \]
- The Kelvin thermometer reading of 233 K is slightly off, but for the purpose of this problem, we will consider the Celsius thermometer as the reference.

4. **Convert Celsius to Fahrenheit:**
- The formula to convert Celsius to Fahrenheit is:
\[ T(°F) = \left( T(°C) \times \frac{9}{5} \right) + 32 \]
- Substituting -40°C into this formula:
\[ T(°F) = \left( -40 \times \frac{9}{5} \right) + 32 = -72 + 32 = -40 °F \]
- Thus, the Fahrenheit thermometer must read -40°F.

5. **Evaluating the Statements:**
- **Statement A:** "Kelvin thermometer should read -233 K" - This is incorrect.
- **Statement B:** "Kelvin thermometer should read -313 K" - This is also incorrect.
- **Statement C:** "The Fahrenheit thermometer must read -40°F" - This is correct.
- **Statement D:** "If water were found in the box, it must be in the liquid state" - This is incorrect since water would be frozen at -40°C.

### Conclusion:
The only necessarily true statement is **Statement C**: "The Fahrenheit thermometer must read -40°F."