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Is there a "good" function $f$ that models the temperature of the Earth's crust/mantle/core measured $r$ miles from the center of the earth? By good, I mean a function that strikes a nice balance between being accuracy and being presentable—easy on the eyes. My intent is to use such a function in an activity for my precalculus class. I'm worried such a function doesn't exist, or would be inherently inaccurate because the temperature isn't sufficiently uniform at a fixed distance from the center.

If no such good function exists, we should list the best we've got.

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No. The earth's composition is stratified, and the different strata have different thermal properties.

You're much better off representing each stratum with an independent function and imposing continuity at the boundaries.

By way of comparison, think of trying to express as a single function the hourly distance traveled of a person as they travel by various different means - on foot, by car, by train, by jet. It's in principle possible, but practically pointless. A system of equations with domains and continuity conditions will represent reality much more effectively.

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  • $\begingroup$ Are there "nice" models for each stratum? It's practically pointless sure, but I don't need practicality here; I'm just designing an activity for a precal class. $\endgroup$ Mar 31, 2022 at 13:17
  • $\begingroup$ @MikePierce yes, on a per stratum basis you can get a close approximation with just a linear function of temperature vs depth, imposing continuity of T(r) across the boundaries and accepting that derivatives will have non physical discontinuities in the model. $\endgroup$
    – g s
    Mar 31, 2022 at 20:02

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