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

Answer:

The formula for the average temperature in the ball is $\bar T = \frac{2\cdot k\cdot R^{1/2}}{3}$. The limit of the average temperature for the ball does not exist.

Step-by-step explanation:

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

$T \propto \sqrt{r}$

$T = k\cdot \sqrt{r}$ (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:

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

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

$\bar T = \frac{k}{R}\cdot \int\limits^{R}_{0} {\sqrt{r}} \, dr$

$\bar T = \frac{2\cdot k}{3\cdot R}\cdot (R^{3/2})$

$\bar T = \frac{2\cdot k\cdot R^{1/2}}{3}$

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 ($n = \frac{1}{2}$) is greater than the maximum exponent of the denominator ($n = 0$). Therefore, the limit of the average temperature for the ball does not exist.