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?

 

Asked on by islnds

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lfryerda | High School Teacher | (Level 2) Educator

Posted on

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.

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