The average temperature for the first four days of a week is `40.2^(@)C` and that of the last four days is `41.3^(@)C`. If the average temperature for the whole week is `40.6^(@)C`, then the temperature on the fourth day is

Options

A. a) `40.8^(@)C`
B. b) `38.5^(@)C`
C. c) `41.3^(@)C`
D. d) `41.8^(@)C`

Answer

To solve the problem, we need to find the temperature on the fourth day of the week given the average temperatures for the first four days, the last four days, and the entire week.

### Step-by-Step Solution:

1. **Calculate the total temperature for the first four days.**
- The average temperature for the first four days is given as \(40.2^\circ C\).
- To find the total temperature for these four days, we multiply the average by the number of days:
\[ \text{Total for first 4 days} = 40.2 \times 4 = 160.8^\circ C \]

2. **Calculate the total temperature for the last four days.**
- The average temperature for the last four days is given as \(41.3^\circ C\).
- Similarly, we calculate the total temperature for these four days:
\[ \text{Total for last 4 days} = 41.3 \times 4 = 165.2^\circ C \]

3. **Calculate the total temperature for the entire week.**
- The average temperature for the whole week is given as \(40.6^\circ C\).
- The total temperature for the seven days of the week is:
\[ \text{Total for the week} = 40.6 \times 7 = 284.2^\circ C \]

4. **Set up the equation to find the temperature on the fourth day.**
- Let \(T\) be the temperature on the fourth day (which is also counted in both the first and last four days).
- The total temperature for the first four days includes \(T\) and the temperatures of the first three days (let's denote them as \(A\), \(B\), and \(C\)):
\[ A + B + C + T = 160.8 \]
- The total temperature for the last four days includes \(T\) and the temperatures of the last three days (let's denote them as \(D\), \(E\), and \(F\)):
\[ T + D + E + F = 165.2 \]

5. **Combine the equations to eliminate \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\).**
- Adding both equations gives:
\[ (A + B + C + T) + (T + D + E + F) = 160.8 + 165.2 \]
- This simplifies to:
\[ A + B + C + D + E + F + 2T = 326.0 \]

6. **Relate this to the total temperature for the week.**
- We know from step 3 that the total temperature for the week is \(284.2^\circ C\), so we can set up the equation:
\[ A + B + C + D + E + F + T = 284.2 \]

7. **Subtract the second equation from the first.**
- We have:
\[ (A + B + C + D + E + F + 2T) - (A + B + C + D + E + F + T) = 326.0 - 284.2 \]
- This simplifies to:
\[ T = 41.8^\circ C \]

### Final Answer:
The temperature on the fourth day is \(41.8^\circ C\).