SOLUTION: Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling f

Algebra ->  Finance -> SOLUTION: Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling f      Log On


   

Question 1032285: Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by dT/dt=k(T-38) where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.
To the nearest minute, how long does it take for the lemonade to cool to 55°F?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The DE dT%2Fdt=k%28T-38%29+ becomes dT%2F%28T-38%29+=+kdt.
==> ln(T-38) = kt + c.
T(0) = 75 ==> ln37 = c. ==> ln%28%28T-38%29%2F37%29+=+kt.
Now T(30) = 60 ==> ln%28%2860-38%29%2F37%29+=+30k, or %281%2F30%29ln%2822%2F37%29+=+k.
==> ln%28%28T-38%29%2F37%29+=+%281%2F30%29ln%2822%2F37%29t, or ln%28%28T-38%29%2F37%29+=+ln%2822%2F37%29%5E%28t%2F30%29
==> %28T-38%29%2F37+=+%2822%2F37%29%5E%28t%2F30%29, or
T+=+38%2B37%2A%2822%2F37%29%5E%28t%2F30%29.
To solve for t when T = 55, plug 55 for T into the last equation.
I will leave the final calculation to you, and you should manage to get highlight%28t+=+45%29 minutes, to the nearest minute.