Question

Suppose the temperature inside a three dimensional ball is proportional to the square root of the distance from the center. Find a formula for the average temperature in the ball. What is the limit of the average temperature as the ball gets larger?

Answer 1

The formula for the average temperature in the ball is
`\bar T = (2\cdot k\cdot R^(1/2))/(3)`. The limit of the average temperature for the ball does not exist.

Explanation:

According to the statement, we have the following direct relationship:

`T \propto √(r)`

`T = k\cdot √(r)` (1)

Where:

`T` - Temperature, measured in degrees Celsius.

`r` - Distance from the center, measured in meters.

`k` - Proportionality constant, measured in degrees Celsius per meter.

Under the assumption that ball is a continuous entity, we find that average temperature in the ball (
`\bar T`), measured in degrees Celsius, is represented by the following integral equation:

`\bar T = (1)/(R)\cdot \int\limits^(R)_(0) {T(r)} \, dr` (2)

By applying (1) in (2), we find that:

`\bar T = (k)/(R)\cdot \int\limits^(R)_(0) {√(r)} \, dr`

`\bar T = (2\cdot k)/(3\cdot R)\cdot (R^(3/2))`

`\bar T = (2\cdot k\cdot R^(1/2))/(3)`

If the ball gets larger, then the limits associated to the average temperature diverges to the infinity as maximum exponent of the numerator of the rational function (
`n = (1)/(2)`) is greater than the maximum exponent of the denominator (
`n = 0`). Therefore, the limit of the average temperature for the ball does not exist.