Min and Max temperature of path in 3D field

In summary, the maximum and minimum temperatures for the given path are located at the limits of t=0 and t=10. Differentiation was used to find the value of t that gives a maximum or minimum temperature, but it was not necessary in this case as the limits of t already gave the correct answer.
  • #1
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Homework Statement


Temperature varies by function T(x,y,z)=3x + 4y + 2z

Path is given by r(t)={
[tex]x(t)=\frac{t^3}{30}+\frac{16t}{9}+7[/tex]
[tex]y(t)=-\frac{t^3}{120}-\frac{13t^2}{30}+28[/tex]
[tex]z(t)=\frac{13t^2}{60}+\frac{4t}{3}-14[/tex]

[tex]t\in \left[0,10\right][/tex]

Question: What is the maximum and minumun temperatures of that path?

Homework Equations



There's a tip in the assignment to use the chain rule.

The Attempt at a Solution


[/B]
[tex]\frac{dT}{dt}=\frac{\partial T}{\partial x}\cdot \frac{\partial x}{\partial t}+\frac{\partial T\:}{\partial \:y}\cdot \:\frac{\partial \:y}{\partial \:t}+\frac{\partial T\:}{\partial \:z}\cdot \:\frac{\partial z}{\partial t}[/tex]

[tex]\frac{dT}{dt}=3\left(\frac{t^2}{10}+\frac{16}{9}\right)+4\left(-\frac{t\left(3t+104\right)}{120}\right)+2\left(\frac{13t}{30}+\frac{4}{3}\right)[/tex]

[tex]\frac{dT}{dt}=\frac{1}{5}\left(t^2-13t+40\right)[/tex]

Then I thought to separate the variables and integrate it to get T, but I end but with constant C.
If I ignore the constant, I end up with wrong answer(min -9/20, max 8)
I cannot think how to continue. I don't know the correct answer either.

Do you think the idea is correct to this point?
Maybe it's something obvious, I might be just baffled with all these derivation steps.

Thank you for reading!
 
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  • #2
I don't understand why you would first differentiate and then "integrate it to get T". You were given T! You seem to have lost track of what you were trying to do.

The reason you differentiated, I hope, was to set the derivative equal to 0 to find the value of t that gives a max or min. Then put whatever value of t you get back into the original equations for x, y, and z.
 
  • #3
Hey, thank you for fast response!

That's exactly what I should do, but for some reason there was so much variables that I didn't realize the obvious course of action.
To my surprise, the correct answer happened to be at the limits of t=0 and t=10 for both, so the resoults from the differentation didn't have any affect.

Thank you for clear explanation!
 
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1. What is the significance of the minimum and maximum temperature of a path in a 3D field?

The minimum and maximum temperature of a path in a 3D field can provide valuable information about the overall temperature distribution in a specific area. This can be useful for predicting weather patterns, understanding the behavior of fluids in a particular region, or identifying areas of potential heat loss or gain in a building or structure.

2. How is the minimum and maximum temperature of a path in a 3D field calculated?

The minimum and maximum temperature of a path in a 3D field is typically calculated by analyzing temperature data from multiple points along the path. This data is then plotted on a graph or mapped onto a 3D model to visualize the temperature distribution and identify the minimum and maximum values.

3. Can the minimum and maximum temperature of a path in a 3D field change over time?

Yes, the minimum and maximum temperature of a path in a 3D field can change over time due to various factors such as changes in weather patterns, variations in environmental conditions, or human activities. It is important to regularly monitor and update temperature data to accurately reflect any changes.

4. How can the minimum and maximum temperature of a path in a 3D field be used for practical applications?

The minimum and maximum temperature of a path in a 3D field can be used for various practical applications such as optimizing heating and cooling systems in buildings, predicting temperature changes in a specific area, or identifying potential areas for energy conservation. It can also be useful for studying the behavior of fluids in industrial processes or natural environments.

5. Are there any limitations to using the minimum and maximum temperature of a path in a 3D field?

Like any scientific data, there are limitations to using the minimum and maximum temperature of a path in a 3D field. These values are based on the accuracy and precision of the temperature measurements taken along the path, and they may not always accurately reflect the true temperature distribution in a specific area. Additionally, other factors such as wind, humidity, and solar radiation can also affect the temperature in a particular region, which may not be accounted for in the minimum and maximum temperature of a path in a 3D field.

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