# A thermometer is taken from a room where the temperature is 18 degrees C to the outdoors, where...

## Question:

A thermometer is taken from a room where the temperature is 18 degrees C to the outdoors, where the temperature is -8 degrees C. After one minute the thermometer reads 5 degrees C.

What will the reading on the thermometer be after 4 more minutes?

When will the thermometer read -7 degrees C?

## Newton's Law of Cooling:

Newton's law of cooling states that " the rate of change of temperature of an object is proportional to the difference of the temperature of the object and the temperature of its surroundings.

According to the Newton's law of cooling, the temperature of a body at time 't' is given by:

{eq}\displaystyle { T_t = T_s + (T_0 - T_s)e^{-kt} } {/eq}

where,

{eq}T_s {/eq} is the temperature of the surroundings

{eq}T_0 {/eq} is the initial temperature

'k' is the constant

## Answer and Explanation: 1

Given:

- Initial Temperature, {eq}\displaystyle { T_0 = 18^{\circ} C } {/eq}

- Surrounding Temperature, {eq}\displaystyle { T_s = - 8^{\circ} C } {/eq}

- After 1 minute that is 60 seconds the temperature is, {eq}\displaystyle { T_{60} = 5^{\circ} C } {/eq}

According to the Newton's law of cooling the temperature at time 't' is given by:

{eq}\displaystyle { T_t = T_s + (T_0 - T_s)e^{-kt} } {/eq}

where,

'k' is constant.

{eq}\displaystyle { \Rightarrow 5 = -8 + (18-(-8)) e^{-60k} \\ \Rightarrow \frac{13}{26} = e^{-60k} \\ \Rightarrow \ln ( \frac{1}{2} ) = \ln e^{-60k} \\ \Rightarrow -0.693 = - 60k \\ \Rightarrow k = 0.01155 } {/eq}

**(1)**

After 4 minutes that is 240 seconds, the temperature would be,

{eq}\displaystyle { T_{240} = -8 + (18-(-8)) e^{-240* 0.01155} \\ = -8 + 26( 0.0625 ) \\ = -6.375^{\circ} C } {/eq}

**(2)**

Time at which the thermometer reading is \ {eq}-7^{\circ} C {/eq}:

{eq}\displaystyle { -7 = -8 + (18-(-8)) e^{-x* 0.01155} \\ \Rightarrow \frac{1}{26} = e^{- 0.01155x} \\ \Rightarrow 26 = e^{ 0.01155x} \\ \Rightarrow \ln (26) = \ln e^{ 0.01155x} \\ \Rightarrow 3.258 = 0.01155x \\ \Rightarrow x = \frac{3.258}{0.01155} \\ = 282.077 \ seconds \\ \approx 4 \ minutes \ 42 \ seconds } {/eq}

which is the answer.

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