the temperature of a 10m long metal bar is 15 degrees celcius at one end and 30 degrees celcius 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 hae this temperature.

The average temperature of the bar can be calculated by finding the mean of the two given temperatures.

(15°C + 30°C) / 2 = 45°C / 2 = 22.5°C

So, the average temperature of the bar is 22.5°C.

At least one place on the bar must have this temperature because of the Intermediate Value Theorem, which states that if a continuous function takes on two values a and b at two distinct points, then it must take on any value between a and b at least once. In this case, the temperature function increases linearly from 15°C to 30°C, so it must take on any intermediate value, including the average temperature of 22.5°C, at least once along the length of the bar.

To find the average temperature of the metal bar, we can take the average of the temperatures at the two ends, as the temperature increases linearly from one end to the other.

Average temperature = (Temperature at one end + Temperature at the other end) / 2

= (15 + 30) / 2 = 45 / 2 = 22.5 degrees Celsius

Now let's consider why at least one place on the bar must have this temperature. Since the temperature increases linearly from the cooler end to the hotter end, the temperature at any point on the bar lies between the temperatures at the two ends (15 degrees Celsius and 30 degrees Celsius).

Therefore, according to the Intermediate Value Theorem, there must be at least one point on the bar where the temperature is equal to the average temperature (22.5 degrees Celsius). This is because the temperature continuously changes as we move along the bar, so it must pass through the average temperature at some point.

To find the average temperature of the bar, we need to consider the fact that the temperature increases linearly from the cooler end to the hotter end. In this case, the temperature gradient can be represented as a straight line.

We can use the equation for a straight line, which is y = mx + b, where y is the temperature, x is the distance from the cooler end, m is the slope of the line (temperature gradient), and b is the y-intercept (initial temperature).

In this scenario, the cooler end has a temperature of 15 degrees Celsius (y1) and is located at x1 = 0m, while the hotter end has a temperature of 30 degrees Celsius (y2) and is located at x2 = 10m.

Using these values, we can calculate the slope (m) of the line:

m = (y2 - y1) / (x2 - x1)
= (30 - 15) / (10 - 0)
= 15 / 10
= 1.5 degrees Celsius per meter

Now we can find the y-intercept (b) by substituting the values of one of the points (e.g., x = 0, y = 15) and the slope (m) into the equation:

15 = 1.5 * 0 + b
15 = 0 + b
b = 15 degrees Celsius

Now that we have the equation of the straight line, we can find the average temperature (y_average) by calculating the integral of the temperature function over the entire length of the bar (x1 to x2):

y_average = (1 / (x2 - x1)) * Integral[ (m * x + b), x1, x2]

Substituting the values into the equation:

y_average = (1 / (10 - 0)) * Integral[ (1.5 * x + 15), 0, 10]

Evaluating the integral:

y_average = (1 / 10) * [ (0.75 * x^2) + (15 * x) ] | from 0 to 10
= (1 / 10) * [ (0.75 * 10^2) + (15 * 10) - (0.75 * 0^2) - (15 * 0) ]
= (1 / 10) * [ 75 + 150 - 0 - 0 ]
= (1 / 10) * 225
= 22.5 degrees Celsius

Therefore, the average temperature of the bar is 22.5 degrees Celsius.

At least one place on the bar must have this temperature because the temperature increases linearly from one end to the other. The average temperature lies between the temperatures at the two ends, but since the temperature increases linearly, there will be a point on the bar where the temperature is equal to the average temperature.