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Mr. Currie pours himself some coffee into a paper cup before making his way to the amusement park. The coffee temperature is 350C when the cup is placed on the kitchen counter with room temperature of 20o C. Mr. Currie was called to the phone for last minute arrangements, and his coffee is forgotten. When he finally returns to his coffee 35 minutes later, the temperature had been decreasing by 1.2% per minute.

a) What type of function best models the cooling of a hot liquid?

b) What is the mathematical model for this situation? (i.e. - the equation)

c) If the optimal temperature for drinking a hot liquid is 280C, at what time would Mr. Currie have had to return in order to enjoy his cup of coffee

According to the Newton's Law of Cooling we have that:

T(t) = Ts + (To - Ts)*e^(-k*t) ;

where:
t is the time in the preferred units (seconds, minutes, hours, etc.)
T(t) is the temperature of the object at time t
Ts is the sorrounding constant temperature
To is the initial temperature of the object
k is a constant to be found

so let's find k

Before I go any further, there is something very wrong with your data
since water boils at 100° C, there is no way that the coffee could have a temperature of 350°
nor can you drink coffee that has a temp of 280°, it would be very hot steam.

its 35° and 20° sorry :\$

Looking at this question again, I think they want this done in a much simpler way.

when t = 35, the T(35) the temperature has decreased by 1.2% each minute,
so temp after 35 minutes = 35(1.015^-35
= 20.7853°

to have a temp of 28°
we need
35(1.015)^-t = 28
1.015^-t = .8
take log of both sides

log (1.015^-t) = log .8
-t = log .8/log 1.015
-t = -1498
t = appr 15

let me know if we have to use Newton's Law of Cooking equation.

where did you get 1.015 from in the first equation?

also what kind of function is this?

sorry, that was a typo, should have been 1.012
for 1 + 1.2%

I am getting myself all messed up here, I noticed that I used 1.015 several times instead of 1.012

Almost too deep into the mess to recover, I should have done this:
reduce by 1.2% per minute ---- >1 - .012 = .988
35(.988)^35 = 22.9°

when does 35° become 28° ?

35(.988)^t = 28
.988^t = 28/35 = .8

t log .988 = log .8
t = log .8/log .988 = 18.48 minutes

does that make more sense?

You are dealing with an "exponential function"