Understanding the Average Temperature Problem

This question asks us to find the highest possible temperature that could have been recorded at noontime on any single day of a week. We are given the average noontime temperature for the entire week (Monday to Sunday) and the lowest temperature recorded during that week.

Calculating the Total Temperature Sum

The average temperature represents the total sum of temperatures divided by the number of days. We know:

• Average noontime temperature = \(31^\circ C\)
• Number of days = 7 (from Monday to Sunday)

To calculate the sum of all temperatures recorded over the 7 days, we multiply the average temperature by the number of days:

Total Sum of Temperatures = Average Temperature \(\times\) Number of Days

Total Sum of Temperatures = \(31^\circ C \times 7\)

Total Sum of Temperatures = \(217^\circ C\)

This means the sum of the noontime temperatures recorded from Monday to Sunday must equal \(217^\circ C\).

Determining the Maximum Possible Temperature

Our goal is to find the maximum possible temperature for one specific day. To make one day's temperature as high as possible, the temperatures on the remaining days must be as low as possible.

The problem states that the lowest temperature recorded during the week was \(30^\circ C\). This means no day had a temperature below \(30^\circ C\), and at least one day had exactly \(30^\circ C\).

Let the temperatures for the 7 days be \(T_1, T_2, T_3, T_4, T_5, T_6, T_7\). We have the equation:

To maximize the temperature on one day (let's call it \(T_{max}\)), we assume the other 6 days had the minimum possible temperature. The minimum allowed temperature is \(30^\circ C\). Therefore, we set the temperatures for the first 6 days to this minimum value:

Now, we calculate the sum of temperatures for these 6 days:

Sum of 6 minimum temperatures = \(6 \times 30^\circ C = 180^\circ C\)

To find the maximum possible temperature (\(T_{max}\)) for the seventh day, we subtract the sum of the 6 minimum temperatures from the total sum of temperatures for the week:

\(T_{max} = (\text{Total Sum of Temperatures}) - (\text{Sum of 6 minimum temperatures})\)

\(T_{max} = 217^\circ C - 180^\circ C\)

\(T_{max} = 37^\circ C\)

Conclusion

Therefore, the maximum possible temperature that could have been recorded at noontime on any one of the days is \(37^\circ C\). This scenario satisfies the condition that the lowest temperature recorded was \(30^\circ C\). We can verify this: if 6 days were \(30^\circ C\) and one day was \(37^\circ C\), the total sum is \((6 \times 30^\circ C) + 37^\circ C = 180^\circ C + 37^\circ C = 217^\circ C\). The average is \(217^\circ C / 7 = 31^\circ C\), which matches the information given in the question.