Question

A cup of coffee is poured at 190F, and is allowed to cool in a 70F room. The temperature of the coffee is 170F after half an hour. How long will it take thte coffee to cool to 120?

Answer 1

exponential decay?

Answer 2

190 = ae^kt + 70
190 = a + 170
a = 120

170 = 120e^k30 + 70
100 = 120e^k30
100/120 = e^30k
ln(100/120) = 30k
ln (100/120)/30 is approx -.0060773852 (do I need to have that exact of a value?)

Answer 3

lebron james

Answer 4

Newton's law of cooling

Answer 5

no wait its kobe bryant

Answer 6

So then I got T(t) = 120e^-0.0060773852 + 70
120 = 120e^-0.0060773852t+70
50/120 = 120 e^-0.0060773852t
ln(50/120) / -0.0060773852 is approx 144 minutes

Answer 7

kevin durant

Answer 8

@batman35, can you stop replying? you are not helping

Answer 9

does it look right?

Answer 10

i'll look it over..

Answer 11

shouldn't it be like this?
p(t) = p_o * e^(-kt)+70
170 = 190*e^(-k*.5)+70

Answer 12

on which part?

Answer 13

to find 'k'

Answer 14

yeah, I wasn't sure if I had the right numbers

Answer 15

i'm not sure what newton's law of cooling is, but i am familiar with exponential decay..

Answer 16

how does that change my k value?

Answer 17

190 would be the initial temperature of the object

Answer 18

@ParthKohli @Hero

Answer 19

@campbell_st @SolomonZelman

Answer 20

@jtryon while DemolisionWolf is helping, please don't tag others. Please respect the helper who is trying helping you.

Answer 21

I only did that because it seems like he wasn't sure

Answer 22

Do I wait for the helper to tell me that he doesn't know and then I can tag others?

Answer 23

hold on, so what is your quesiton? are u asking if you did it correct?

Answer 24

I'm asking if I did it right or I put the numbers in the wrong places because I don't know if 144 minutes is right

Answer 25

see how you started here:

190 = ae^kt + 70
a actually means beginning temperature, so its like this:
T(t) = 190 e^(-k*t) + 70

-k because it is decaying, positive k would be increasing

Answer 26

so that is going to change my answer

Answer 27

but don't you plug in the value for a so you have 170 = 120e^k30+70?

Answer 28

@DemolisionWolf

Answer 29

the equation of the Newton's cooling Law
\[T(t) = T_{room}+ (T_{initial}-T_{room})e^{-kt}\]
so that your equation is
\[T(t) = 70 + 120e^{-kt}\]
to me, your work is ok until it
Now, after 30 minutes, the temperature of the coffee is 170
It means T(30) = 170, plug into the formula to find k
\[T(30) = 70 +120 e^{-30k}= 170\\100 = 120 e^{-30k}\\\dfrac{100}{120}=e^{-30k} =\dfrac{10}{12}\]
\[e^{30k}=\dfrac{12}{10}\rightarrow 30k = ln\dfrac {12}{10}\rightarrow k = 0.006\]
replug into the formula
\[T(t) = 70+120e^{-0.006t}\]
we need to find out how long will it take to cool it down to 120
that means \[120 = 70 +120e^{-0.006t}\\\frac{50}{120}=e^{-0.029t}\rightarrow ln\dfrac{12}{5}=0.006t \\t =145.9 ~min\]
that's all I know.

Answer 30

@Loser, where did you get -0.029t?

Answer 31

I see where you got -0.006t

Answer 32

should it not be 0.006 instead of -0.006? I am trying to follow along with the way that you worked out the problem

Answer 33

because k -> 0.006

Answer 34

nope, typo, not 0.029, it is0.006

Answer 35

and it is not -0.006 but 0.006?

Answer 36

so you get ln (12/5) / .006 is approximately 146 mins

Answer 37

don't know why you put -30k and - as you were going through it

Answer 38

yup

Answer 39

the constants to me seem like they are +30k and +0.006

Answer 40

Actually, I made a short research to know what the Newton Cooling law is and worked on your problem as above. ( since I saw no one helped you hihihihi) However, I don't think I make any mistake except the typo as corrected. Hopefully it is correct. Good luck

Answer 41

It is correct. I was just trying to figure out where -30k came from since 30 represents the time and it is positive