Question

Hi, I kinda have a trouble with exponantial growth/decay.

example 1 works just fine
example 2 - have no Idea how to sole it

example 1:
some one invests 1000 USD at x%. After 10 years the investment sum doubes find x

solution 2000=1000(1+x)^10

example 2:
A bottle of soda pop at room temperature (72 deg) is placed in a refrigerator where
the temperature is 44 deg After half an hour the soda pop has cooled to 61 deg.
How long does it take for the soda pop to cool to 50 deg?

how to find a solution for example 2???

Answer 1

its an exponential function, and uses Newton's law of cooling, which uses ambient temperature...

so the soda can't cool to a temperature below the ambient temperature of 44

and you know T(30) = 61

so the model is basically

\[T = T_{a} + (T_{o} - T_{a}) e^{-kt}\]

Ta is the ambient temperature and To is the initial temperature

so you have

\[T = 44 +(72 - 44) e^{-kt}\]

or

\[T = 44 + 28e^{-kt}\]

so what you need to do is find the constant of decay k

using the temperature of 61 then t = 30

so

\[61 = 44 + 28e^{-k \times 30}\]

or

\[17 = 28e^{-k \times 30}\]

divide by 28

\[\frac{17}{28} = e^{-k \times 30}\]

take the base e log of both sides

\[\ln(\frac{17}{28})= - 30k\]

divide both sides by -30

and you'll have the decay constant

k = 0.016633

so now your decay model is

\[T = 44 + 28e^{-0.16633t}\]

for the 2nd part of the question you'll need to substitute T = 50 and then solve for t.

here is a link I googled that may help

http://www.sosmath.com/diffeq/first/application/newton/newton.html

Answer 2

does the Newton's law of cooling mean that the rate of cooling k is not constant?

Answer 3

the rate of cooling is proportional to the original function and the constant of decay k is a constant value..

read the link... it may make it clearer..

Answer 4

ok. than k is a constant. thats clear.
the more the difference between the temp. of a body and the temp. of the environtment, the greater k is. thats also clear.

to find k the newtonian law should be used: k=-1/30*ln(17/28).
than to find the temp. change over time the formula from example 1 can be used.

Answer 5

thats correct...

and to find out how long it takes the temp to get to 50 uses the same k value

and then you are solving for t

in a similar way to finding k.

hope it helps

Answer 6

yes thanks.