Question

Can someone please help me with this question? ;]
http://puu.sh/7oHCi.png

Answer 1

do you know how to integrate?

Answer 2

yes I do

Answer 3

but how do you set this up? :o

Answer 4

The rate at which the temperature of the coffee changes with time at any instant is proportional to the difference between the temperature of the coffee at that instant and the (assumed constant) room temperature (25 °C here).

Since H(0) = 100 > 25, the temperature decreases with time, so dH/dt < 0. Therefore, to make the right side of the DE to be negative for, say, t = 0, we must have k > 0. The minus sign in front of the k in the DE is a reminder to us that dH/dt < 0. You could very well leave it out, and you would still get the right answer (it would be, of course, the negative of the result you get by explicitly putting it in).

dH/dt = -kH + 25k
dH/dt + kH = 25k

This is first-order linear, and so has the integrating factor e^( ∫k dt) = e^(kt)

e^(kt)[ dH/dt + kH ] = 25ke^(kt)
d/dt [ H e^(kt) ] = 25ke^(kt)

Integrating,

H(t) e^(kt) = 25 e^(kt) + C
H(t) = 25 + Ce^(-kt)
H(0) = 100 ⇒ C = 75
H(t) = 25 + 75e^(-kt)

Using the given condition,

H(3) = 90 ⇒ 25 + 75e^(-3k) = 90

You can do the rest. You should get 18 minutes for the last question. (18.077055 to six decimals).

Answer 5

Welcome to Openstudy @Sultillort47