A temperature of a 10m long metal bar is 15C at one end and 30C on the other end. Assuming that the temperature increases linearly from the cooler end to the hotter end, what is the average temperature of the bar? Also, explain why at least one place on the bar must have this temperature.
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To find the average temperature of the metal bar, we first need to find the temperature at the midpoint of the bar, as the temperature is assumed to increase linearly.
The temperature difference between the two ends of the bar is 30C - 15C = 15C.
Since the length of the bar is 10m, the temperature difference per unit length is 15C / 10m = 1.5C/m.
The midpoint of the bar is located 5m away from both ends. So, at the midpoint, the temperature increase from the cooler end is 1.5C/m * 5m = 7.5C.
Therefore, the average temperature of the bar is the sum of the initial temperature at one end (15C) and the temperature increase at the midpoint (7.5C), divided by 2:
Average temperature = (15C + 7.5C) / 2 = 22.5C.
Now, let's explain why at least one place on the bar must have this average temperature.
Since the temperature increases linearly from the cooler end to the hotter end, the temperature distribution along the bar forms a straight line. This linear relationship means that the temperature changes uniformly throughout the bar.
At the midpoint of the bar, the temperature increase is exactly half of the total temperature difference. In this case, the temperature increase at the midpoint is 7.5C, which is half of the total temperature difference of 15C.
Therefore, at the midpoint of the bar, the temperature is exactly the average temperature of the whole bar, which is 22.5C.
In conclusion, at least one place on the bar must have the average temperature, which in this case is at the midpoint.