Question 1165586: According to Newton’s law of cooling, the temperature of an object changes at a rate proportional to the difference in
temperature between the object and the outside medium. If an object whose temperature is 70OF is placed in a medium whose
temperature is 20O and is found to be 40O after 3 minutes, what will its temperature be after 6 minutes?
a. 25OF b. 28OF c. 31OF d. 34OF
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe your answer will be 280.
the formula i used is f = p * (1 + r) ^ n
this formula will work off the difference between the temperature of the object and the temperature of the medium.
in the beginning, the difference is 700 - 200 = 500
after 3 minutes, the difference is 400 - 200 = 200
the formula becomes:
200 = 500 * (1 + r) ^ 3
solve for r to get:
r = (200/500) ^ (1/3) - 1
that gets you:
r = -.2631957003.
the formula becomes:
200 = 500 * (1 - .2631957003) ^ 3.
evaluate the equation to get:
200 = 200
this confirms the rate is correct.
after 6 minutes, the formula becomes:
f = 500 * (1 - .2631957003) ^ 6.
solve for f to get:
f = 80.
the difference is 80 after 6 minutes.
since the temperature of the medium is assumed to not change, that means the temperature of the object is 200 + 80 = 280 degrees.
note that, if the object was placed in the medium forever, then the assumption is that the object will becomes 200 degrees, same as the medium.
you can graph the function.
it looks like this.
to test the claim that eventually it would become 200 degrees, make n a very large number and solve for that.
for example:
f = 500 * (1 - .2631937003) ^ 30 gets you:
f = .0524288.
the temperature of the object would then be 200.0524288 degrees.
that's pretty close to 200.
make n larger and it will get closer and closer to 200.
for example, when n = 60, f = .000005
round to anything less than 5 decimal digits and the difference is 0.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website!
According to Newton’s law of cooling, the temperature of an object changes at a rate proportional to the difference in temperature between the object and the outside medium. If an object whose temperature is 70OF is placed in a medium whose temperature is 20O and is found to be 40O after 3 minutes, what will its temperature be after 6 minutes?
a. 25OF b. 28OF c. 31OF d. 34OF
Newton's law of cooling formula:  where:  = time (t) at a COOLED temperature (3 minutes, in this case)
 = TEMPERATURE (T) at a given time (t) (400oF, in this case)
 = SURROUNDING temperature (200oF, in this case)
 = ORIGINAL/INITIAL temperature (700oF, in this case)
 = CONSTANT or COOLING rate (Unknown, in this case)
In this case, we 1st have to determine the value of k, and so: 
, where: = time (t) at a COOLED temperature (6 minutes, in this case)
= TEMPERATURE (T) at a given time (t) (Unknown, in this case)
= SURROUNDING temperature (200oF, in this case)
= ORIGINAL/INITIAL temperature (700oF, in this case)
= CONSTANT or COOLING rate ( , in this case)
Temperature, 6 minutes after, or 
|
|
|
