# Application of first order differential equation-tomorrow test

• May 29th 2009, 09:17 AM
sanikui
Application of first order differential equation-tomorrow test
I have a problem of this type of question.
An object thrown into a large body of water cools at a rate proportional to the difference between its temperature and the water temperature. Suppose know that the the water is at a temperature of 27 degrees Celsius. After 4 minutes the object's temperature is 67 degrees, and after 9 minutes the object's temperature is 47 degrees Celsius. What was the temperature of the object when it was thrown into the water?

Differential equation is dQ/dt = -k(Q-Qs)

Q is the temperature of an object
Qs is the water temperature.

Not like other question, i can find K (constant) but in this question K ( constant ) i even cant find it out.
• May 29th 2009, 10:05 AM
skeeter
Quote:

Originally Posted by sanikui
I have a problem of this type of question.
An object thrown into a large body of water cools at a rate proportional to the difference between its temperature and the water temperature. Suppose know that the the water is at a temperature of 27 degrees Celsius. After 4 minutes the object's temperature is 67 degrees, and after 9 minutes the object's temperature is 47 degrees Celsius. What was the temperature of the object when it was thrown into the water?

Differential equation is dQ/dt = -k(Q-Qs)

Q is the temperature of an object
Qs is the water temperature.

Not like other question, i can find K (constant) but in this question K ( constant ) i even cant find it out.

$\frac{dQ}{dt} = k(Q - 27)$

$\frac{dQ}{Q-27} = k \, dt$

$\ln(Q-27) = kt + C$

$
Q = 27 + Ae^{kt}
$

at $t = 4$, $Q = 67$

$67 = 27 + Ae^{4k}$

$
40 = Ae^{4k}
$

at $t = 9$, $Q = 47$

$
47 = 27 + Ae^{9k}
$

$
20 = Ae^{9k}
$

$\frac{40}{20} = \frac{Ae^{4k}}{Ae^{9k}}$

$
2 = e^{-5k}
$

solve for k , then solve for A and finish.
• May 29th 2009, 10:09 AM
Danneedshelp
haha, too fast for me. I just finished the problem. The work above is correct...or at least the same as what I got.
• May 29th 2009, 10:19 AM
sanikui
Quote:

Originally Posted by skeeter
$\frac{dQ}{dt} = k(Q - 27)$

$\frac{dQ}{Q-27} = k \, dt$

$\ln(Q-27) = kt + C$

$
Q = 27 + Ae^{kt}
$

at $t = 4$, $Q = 67$

$67 = 27 + Ae^{4k}$

$
40 = Ae^{4k}
$

at $t = 9$, $Q = 47$

$
47 = 27 + Ae^{9k}
$

$
20 = Ae^{9k}
$

$\frac{40}{20} = \frac{Ae^{4k}}{Ae^{9k}}$

$
2 = e^{-5k}
$

solve for k , then solve for A and finish.

Okay, i get it...hope tomorrow i can do this type of question.
Thanks everyone. V^^