Mathematics, 18.11.2020 16:40 accis

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?

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Answer from: cat20220

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:

- Temperature, measured in degrees Celsius.

- Distance from the center, measured in meters.

- 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.

Answer from: Quest

step-by-step explanation:

a. since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since 2010 0 1 2 3 4 5

profits made in 1000's 39 84 111 120 111 84

we find that profit gradually increases form 2010 to 2013 upto 120, then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a. since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

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