Math Is Fun Forum

bobbym - Yes, I meant those things. I am more interested in whether or not I am correct. And I do not have Stewart's calculus book. Do they get the same answer?

bob bundy - I do not know for sure. But yes, the answer seems reasonable, especially for how slowly the roads are getting plowed. It must have been snowing for quite some time before they got started at 8am.

Finished. But I do not know if I am correct. I have 1:42 AM.

What are you guys coming up with?

I'll post my work so you can follow how I came to this solution.

Hi... sorry, distracted with other matters. Let me re-read the problem.

I would start by forming two differential equations. One to describe the rate the snow is accumulating, such as

where the rate of the snowfall is constant, and then another equation to describe the velocity of the plow:

where the rate of the plow is inversely proportional to the amount of snow on the ground.

Solving the first differential equation we get

where D is a new constant. Substituting this into the second differential equation for S we then have

And solving this differential equation yields:

But now we have four parameters - k, C, D, and E - along with our variables t and P and it needs to be simplified in some manner so as to lessen the parameters by at least 1 because we have two values for time and progress and our third equation in such a system would want to solve for t when the snow fall is 0. Or something like that.

How about

I think this could be used to create a system of equations using time t and p(t) for different values to solve for A, B, and E. Am I going in the right direction, do you think?

This conversation was amusing to come home to.

Thanks for your help guys.

I see my mistake. Thanks...

That is what I did at first and I got some crazy answer like negative 27 hours. Here it is worked out:

Now I am getting t = -0.052575

Which is nonsense. Unless it means the event occurred minutes beforehand.

Horray. Here is my work:

Solving first for C with the initial condition T(0) = 45 which is where T_s is 120 and where the plasma is 40 F...

Therefore, the equation is now

Solving for the proportionality constant k the initial conditions are used again:

So the new function solved for C and k is

The new function is in terms of T_s and t and so to calculate how long it will take in time (t) for the plasma to warm to 90 degrees (T(t)) the equation can be solved for t:

And by that t comes out to be approximately 95 minutes.

Thanks for the help. I now see how solving for c and then k using the initial conditions allows for the establishment of a function with which to calculate new values for time.

Less than two years ago I hardly knew much more than how to solve 3x = 6 for x. In that time I have gotten as far as I have and somehow, someway, in all that time I never realized a negative sign made equal the reciprocal of the inside of a logarithm. How did that ever escape me? And for so long?

You learn new stuff every day.

Finishing up the problem now...

Awesome! Thanks! That is going into my notebook for sure.

Let me the problem that way. Hang on.

t = 2? That isn't right. I must have messed something up.

Because yours

and mine

have the order of the temperature of the object and the temperature of the surrounding area reversed. Which one is the proper order?