Question

PLEASE HELP WITH WORD PROBLEM???
A cup of coffee was made at a temperature of 90C and cools according to Newton's law of cooling. The room temperature is 30C. If the temperature of the coffee 20 minutes after being made was 40 C
When was the temperature of the coffee 80C Time = how many minutes??

Answer 1

Is the formula something like

\[T_{n} = T_{o} + e^{kt}\]

Answer 2

i believe so i know after 10 minutes the temp. is 54 degrees

Answer 3

something is wrong in the formula... have you been given a formula for this..?

Answer 4

hold on

Answer 5

i haven't been given a formula for this..

Answer 6

ok... I've just done some reading online it looks like

\[T = e^{kt + C} + 30\]

so start by finding C, which uses the initial values t= 0 and T = 90

Answer 7

hold on let me check

Answer 8

for some reason its not correct

Answer 9

yes

Answer 10

oops... found another mistake... if you give me a minute I'll correct everything...

Answer 11

ok thanks!

Answer 12

ok so the model is

\[T = e^{kt + C} + 30\]

start by using the initial conditions t = 0 and T = 90 to find C

\[C = \ln(90 -30)\]

so

C = 4.09434

the model now becomes

\[T = e^{kt + 4.09434} + 30\]

next use t = 20 and T = 40 to find the value of k

\[40 = e^{20k + 4.09434} + 30\]

which simplifies to

\[k = \frac{\ln(10) - 4.09434}{20}\]

so

k = -0.089588

so the final model is

\[T = e^{-0.089588t + 4.09434} + 30\]

now you need to find t, when T = 80

\[80 = e^{-0.089588 + 4.09434} + 30\]

solving for t you get

\[t = \frac{\ln(50) - 4.09434}{-0.089588}\]

so t = 2.03 mins

or the coffee cools to 80 in 2.03 mins

Answer 13

I can't tell from the question is your answer should be 2 mins or 2.03 mins..

Answer 14

2.03 is correct, your sooo amazing!!! :) Can you help me with another question??

Answer 15

lol... sorry I have to go to school....