Math Is Fun Forum

A heating a cooling differential equation word problem:

"Blood plasma is stored at 40 degrees F. Before it can be used, it must be at 90 degrees F. When the plasma is placed in an oven at 120 degrees F, it takes 45 minutes for the plasma to warm to 90 degrees F. Assume Newton's Law of cooling applies. How long will it take the plasma to warm to 90 degrees F if the oven temperature is set at 100 degrees F?"

Looking up Newton's Law of Cooling, I believe a good starting place is to state the equation as

where T is the temperature at time t and that change is directly proportional to the difference between the surrounding temperature of the chamber and that of the object.

Listing the initial values...

And so, what the problem is asking for is for us to find the time it will take to warm the plasma all the way up to 90 F if the oven is only 100. Just working off of the definition of Newton's Law of Cooling, here is a first attempt at setting up the problem:

That isn't right but I am not sure how to specify two different temperatures for two different objects, the oven and the plasma, t 0 and time t.

I have never taken physics and I do not understand what the question is asking.

"Kirchoff's Law states that a simple circuit containing a resistor of R ohms and an inductor of L Henrys in series with a source of electromotive force that supplies a voltage E(t) volts at time t satisfies

where I is the current measured in amperes.

Consider a circuit with L = 6 Henrys, R = 6 ohms, and a battery supplying a constant 12 volts. If I = 0 at t = 0, find I at time t."

I have no idea what most of that means. Do I need to understand physics to be able to solve the problem? Or is it like that raindrop problem from a week or so back that implies physics but does not require a knowledge of it?

Thanks!

Hello.

I am still getting the hang of mixture problems. Here is one.

"A mixing tank initially contains 120 gallons of brine which in turn contains 75 pounds of salt in its solution. A new brine containing 1.2 pounds of salt per gallon begins entering the tank at t=0 and at the rate of 2 gallons per minute while the uniform mixture flows out of the tank at 1 gallon per minute. Assuming the mixture is kept always uniform, find the amount of salt in the tank at the end of 1 hour."

And here is how I have it set up:

I would like some feed back on how it is set up because, done this way, it is not separable and therefore needs one of those exponentials with an integral in the power.

How's it look? And yep, I am solving it by hand.

Hello!

I am curious and trying to learn more about Thiele's Interpolation Formula, the mechanics of how it works, and understanding, in general, how it is derived, among other things.

That which fascinates me about the formula is an odd inversion of the function within the function itself that seems to appear with any set of three points friendly enough with the function to not divide by zero. For example, if you solve the formula for the three symmetric points

you get the formula

which, on the domain from 0 to 100 creates a pretty graph whose derivatives are 9 when x = 0, 1 when x = 25, and 1/9 when x = 100, simply using the values of x from the points given above.

What's bizarre is that the function has an inverse of itself across the vertical asymptotic line and in the second quadrant which is shifted over by a value equal but negative to the second x value (-25)and up by the second y value (75). The inverse in the second quadrant therefore has derivatives equal to and equidistant to those of the function's curve in the first quadrant, proving they are the same.

How does this work? How can a function contain its own inverse?

My observation of this goings on stems from my attempts at curve fitting which can be found elsewhere in the "Help Me!" portion of the forum, though by now I believe it must be buried several pages back.

Thank you for your input!