Newton's Law Of Cooling question from old New Senior Mathematics 3UNIT (1 Viewer)

McRafi9 · Mar 12, 2017

Ex25(e) page 129 Q4. old New Senior Mathematics 3UNIT

A body, initially at room temperature 20°C, is heated so that its temperature would rise by 5°C/min if no cooling took place. Cooling does occur in accordance with Newton's Law of Cooling and the maximum temperature the body could attain is 120°C. How long would it take to reach a temperature of 100°C?

Please explain like I am not smart. I can do the basic Law of Cooling questions, but I have no idea how to even approach this one.

Mahan1 · Mar 12, 2017

McRafi9 said:
Ex25(e) page 129 Q4. old New Senior Mathematics 3UNIT

A body, initially at room temperature 20°C, is heated so that its temperature would rise by 5°C/min if no cooling took place. Cooling does occur in accordance with Newton's Law of Cooling and the maximum temperature the body could attain is 120°C. How long would it take to reach a temperature of 100°C?

Please explain like I am not smart. I can do the basic Law of Cooling questions, but I have no idea how to even approach this one.

Well
$T_{0} = 20, T_{f} = 120 , k =5$

Using the Newton's law:

$T(t) = T_{f} + Ae^{-5t} \Rightarrow T(0) = 20 = 120 + Ae^{0} \Rightarrow A=-100$

We want to find the time it takes for for the body temp reaches 100, so assume for some t, T(t) =100:

$T(t) = 120 - 100 e^{-5t} =100 \Rightarrow 100 e^{-5t} = 20 \Rightarrow e^{-5t} = \frac{1}{5}$

$\therefore t = \frac{ln(5)}{5}$

McRafi9 · Mar 14, 2017

Mahan1 said:
Well
$T_{0} = 20, T_{f} = 120 , k =5$

Using the Newton's law:

$T(t) = T_{f} + Ae^{-5t} \Rightarrow T(0) = 20 = 120 + Ae^{0} \Rightarrow A=-100$

We want to find the time it takes for for the body temp reaches 100, so assume for some t, T(t) =100:

$T(t) = 120 - 100 e^{-5t} =100 \Rightarrow 100 e^{-5t} = 20 \Rightarrow e^{-5t} = \frac{1}{5}$

$\therefore t = \frac{ln(5)}{5}$

Answers say 32.2 min. ln(5)/5 = 0.322 (3d.p.). Why do we need to multiply by 100?

Newton's Law Of Cooling question from old New Senior Mathematics 3UNIT (1 Viewer)

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