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:

The formula for the average temperature in the ball is [tex]\bar T = \frac{2\cdot k\cdot R^{1/2}}{3}[/tex]. 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:

[tex]T \propto \sqrt{r}[/tex]

[tex]T = k\cdot \sqrt{r}[/tex] (1)

Where:

[tex]T[/tex] - Temperature, measured in degrees Celsius.

[tex]r[/tex] - Distance from the center, measured in meters.

[tex]k[/tex] - 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 ([tex]\bar T[/tex]), measured in degrees Celsius, is represented by the following integral equation:

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

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

[tex]\bar T = \frac{k}{R}\cdot \int\limits^{R}_{0} {\sqrt{r}} \, dr[/tex]

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

[tex]\bar T = \frac{2\cdot k\cdot R^{1/2}}{3}[/tex]

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

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