Mathematical mdelling with diffrential equations

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January 7th 2011, 07:45 AM
hejmh

Mathematical mdelling with diffrential equations

A cold water has a temperature of 10 'c. It stands in the room of the temperature 30 'c. After 10 min the temperature of the water rises to 15 'c. The temperature y 'c,
rises at a rate that is proportional to the difference of 30-y.

a) Setup a diffrential equation that describes this situation.
b)What will be the temperature of water after 20 min.
c)How fast does the temperature rise when it is 20 'c.

I would be grateful if somebody could help me.
January 7th 2011, 08:04 AM
CaptainBlack

Quote:

Originally Posted by hejmh

A cold water has a temperature of 10 'c. It stands in the room of the temperature 30 'c. After 10 min the temperature of the water rises to 15 'c. The temperature y 'c,
rises at a rate that is proportional to the difference of 30-y.

a) Setup a diffrential equation that describes this situation.
b)What will be the temperature of water after 20 min.
c)How fast does the temperature rise when it is 20 'c.

I would be grateful if somebody could help me.

You could start with (a) the last sentence of the problem statement before it "The temperature 'y' in degrees centigrade,
rises at a rate that is proportional to the difference of '30-y' " is what you need to rewrite as a differential equation.

CB
January 7th 2011, 10:27 AM
janvdl

I would use Newton's Law of Cooling on this one.

$\frac{dT}{dt} = k(30 - T)$

Seperate, integrate by parts.

Then solve for the constant using the known temperature $T = 10$ when $t = 0$

Solve for the constant of proportionality, $k$ by solving the equation when $T = 15$ when $t = 10$