# A pan of water is brought to boil and then removed from the heat. Every 5 minutes thereafter the difference between the temperature of the water and the  room tempreture is reduced by 50%. If room...

A pan of water is brought to boil and then removed from the heat. Every 5 minutes thereafter the difference between the temperature of the water and the  room tempreture is reduced by 50%. If room temperature is 20 degree C, Express the temperature of the water as a function of the time since it was removed from the heat? Also how many minutes does it take for the tempreture of the waterto reach 30 degreeC?

lfryerda | Certified Educator

Since the difference in temperature is reduced by 50%, we can model the temperature difference using a half-life function.

`A(t)=A_0(1/2)^{t/k}`  where `A(t)` is the temperature difference after `t` minutes, `A_0` is the initial temperature difference, and `k` is the half-life time.

The half-life time is 5 minutes, so `k=5` .

Although the initial temperature is boiling (100 degrees C), the initial temperature difference is `100-20=80` degrees so `A_0=80` .

The final temperature difference is `30-20=10` so `A(t)=10` .

Now substitute to get:

`10=80(1/2)^{t/5}`   divide by 80

`1/8=(1/2)^{t/5}`   replace left side by power of 1/2.

`(1/2)^3=(1/2)^{t/5}`   compare exponents

`3=t/5`  cross multiply

`t=15`

It will take 15 minutes for the temperature to reach 30 degrees C.