You are taking a road trip in a car without A/C. The temperture in the car is 108 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 41 minutes. Given that T-A/To-A=e^-kt.
where T= the temperature of the pop at time t.
To= the initial temperature of the pop.
A= the temperature in the car.
k= a constant that corresponds to the warming rate.
and t= the length of time that the pop has been warming up. How many minutes will it take the pop to reach 81 degrees F?

Question

You are taking a road trip in a car without A/C. The temperture in the car is 108 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 41 minutes. Given that T-A/To-A=e^-kt. 
where T= the temperature of the pop at time t.
To= the initial temperature of the pop. 
A= the temperature in the car. 
k= a constant that corresponds to the warming rate.
and t= the length of time that the pop has been warming up. How many minutes will it take the pop to reach 81 degrees F?

anonymous · Answer

oh lord really?  we can do this but it is going to take some time

anonymous · Answer

tell me about it. Ive tried it a few already

anonymous · Answer

ok first off forget about the verbiage

anonymous · Answer

ok should we set it up first?

anonymous · Answer

$$108-45=63$$ and 
$$108-60=48$$ those are the numbers we will be working with

anonymous · Answer

we start counting at time t = 0 and 
at t = 0 we have 63
at t = 41 we have 48 
so the problem is find the exponential function containing the points (0, 63) and (41,48)

anonymous · Answer

so we have 
$$y=63e^{rt}$$ and we are looking for r
we know when t = 41 we have y = 48 so write 
$$48=63e^{41r}$$ and solve for r

anonymous · Answer

solving here is a little tricky. should i divide by 63 then multiply both sides by natural log?

anonymous · Answer

right i screwed that up!  you are right except please don't say to your teacher "multiply by the log" he or she will think you are a moron.  say "take the log" and they will be happier.  
write 
$$\frac{48}{63}=e^{41r}$$
$$\ln(\frac{48}{63})=41r$$
$$r=\frac{\ln(\frac{48}{63})}{41}$$

anonymous · Answer

so now we have r and we can compute it with a calculator

anonymous · Answer

its -.0066

anonymous · Answer

-0.006632529

anonymous · Answer

yeah got it. ok now we want to know when the thing will be 81 degrees but BE CAREFUL because it is the difference in the temperatures we are looking at

anonymous · Answer

so the difference is 27?

anonymous · Answer

right.  exactly!

anonymous · Answer

so we want to solve 
$$63e^{-.0066t}=27$$ for t

anonymous · Answer

and we run through the same routine again

anonymous · Answer

divide by 63, take the log, divide by -.0066

radar · Answer

Can you calculate k the warming rate from the fact the pop goes from 45 to 60 (15 degrees) in 41 minutes, or are you not concerned with k??

anonymous · Answer

@radar no, because it is the difference in the temperatures that decays

anonymous · Answer

128.3684

anonymous · Answer

we got k = -.0066 only i called it r .  same thing.

radar · Answer

Oh O.K.

anonymous · Answer

@bbaker i got the same thing.  there is another way to do this that is perhaps quicker but never mind it involves basically the same work

anonymous · Answer

it is not accepting that answer. I don't know why

anonymous · Answer

always work with the differences in the temperature and then you don't have to worry so much about the cumbersome formula
maybe we made a mistake let me check

anonymous · Answer

maybe it mean how many MORE minutes will it take, in which case subtract 41

anonymous · Answer

hmmm. not it either. Im stumped haha

anonymous · Answer

even using more decimal place accuracy i still get 128 minutes rounded

anonymous · Answer

ok. I will have to email my teacher. Thank you

anonymous · Answer

sorry we did not get an answer that the system likes but i don't see a mistake.