Question

Newton's Law of Cooling

A Cup of coffee placed in a room cools from 160F to 150F in one minute. If the room temperature is 65F, Find the temperature in ten minutes.

Answer 1

"Newton's Law of Cooling"\[
T(t) = (T_0-T_r)e^{kt} + T_r
\]Correct me if I'm wrong on this formula.

"A Cup of coffee placed in a room cools from 160F to 150F in one minute."\[
T(0) = T_0 = 160^\circ F \\
T(1) = 150^\circ F
\](This will help us solve for \(k\))

"the room temperature is 65F"\[
T_r = 60 ^\circ F
\]"Find the temperature in ten minutes"\[
T(10) = ?
\]

Answer 2

@zhengcl86 Think you can take it from here?

Answer 3

@wio i have a different formula

Answer 4

Tr=60∘F why is it 60? and not 65?

Answer 5

That is typo... what is your formula?

Answer 6

Xk - Xk-1 = A (Xk-1 - 65)

Answer 7

What?

Answer 8

so i am using somewhat a similar formula then yours but i still get an answer

Answer 9

so i am using a different formula that i found online

T(t) = T0 + ( T1 - T0) e^-k(t)

T(t) = 65 +(160-65)e^-k(t)
T(t) = 65+95 e^-k(t)
time = 1 Temp = 150
150 = 65+95 e^-k(1)
85 = 95 e^-k
.8947368421 = -k
T(t) 65+95 e^-.8947368421(10)
T(10) = 160 - 8.947368421
T = 151

which still makes no sense!! what am i doing wrong?

Answer 10

don't know what i did wrong but the temperature went up 1 degrees in 10 minutes lol

Answer 11

My formula was perfectly fine.

Answer 12

\[\begin{array}{}
T(t) &=& (T_0-T_r)e^{kt} + T_r \\
150 &=& (160-65)e^{k(1)}+65 \\
85 &=& 95e^{k} \\
\frac{17}{19} &=& e^{k} \\
\end{array}
\]Plugging it back in: \[\begin{array}{}
T(t) &=& 95\left(\frac{17}{19}\right)^{t}+65 \\
T(10) &=& 95\left(\frac{17}{19}\right)^{10}+65 \\
&\approx & 96
\end{array}
\]

Answer 13

http://www.wolframalpha.com/input/?i=95%5Cleft%28%5Cfrac%7B17%7D%7B19%7D%5Cright%29%5E%7B10%7D%2B65

Answer 14

how is the work done in T(10)

Answer 15

I plugged it into wolfram

Answer 16

oh.....well thats not gonna help me during the test lol thanks you...

Answer 17

You're not allowed to use a calculator?

Answer 18

Just do the order of operations.

Answer 19

i am, im trying to do the calculations now and its not giving me 96

Answer 20

so 17/19 x 10 = 8.947368421

Answer 21

that multple by 95 is 850

Answer 22

Not multiplication. It's an exponent.

Answer 23

then 850 + 65 is 915

Answer 24

i am still very lost on the last part

Answer 25

Exponents are repeated multipicaton... \[
x^{10} = \underbrace{x\times x \times x \times ... \times x}_{10\text{ times}}
\]

Answer 26

where does T(10) go?

Answer 27

so with the exponent of 17/19 to the 10 is 8.005540166

Answer 28

\[
\left(\frac{17}{19}\right)^{10}\approx 0.329
\] http://www.wolframalpha.com/input/?i=%5Cleft%28%5Cfrac%7B17%7D%7B19%7D%5Cright%29%5E%7B10%7D

Answer 29

dont know why i am not getting that on my calculator

Answer 30

well thank you for all your help