A cold yam is put into a hot oven to bake. The temperature of the yam begins to rise. The rate, R...

Question:

A cold yam is put into a hot oven to bake. The temperature of the yam begins to rise. The rate, {eq}R {/eq} (in degrees per minute), at which the temperature of yam rises is governed by Newton's Law of Heating, which says that the rate is proportional to the temperature different between the yam and the oven. If the oven is at {eq}350 \, ^{\circ} F {/eq} and the temperature of yam is {eq}H \, ^{\circ} F {/eq}, write a formula giving {eq}R {/eq} as a function of {eq}H {/eq}.

Newton's Cooling Law

Newton's cooling Law is an exponential function that describes the change in temperature of an object placed in an environment with constant temperature. The exponential function is given by {eq}T(t) = T_0 + (T_i-T_0)e^{-kt} {/eq}.

Answer and Explanation:


Newton's cooling law or heating law is given by the equation

{eq}T(t) = T_0 + (T_i-T_0)e^{-kt} {/eq}

where T(t) is the temperature of the object at any time, t, {eq}T_0 {/eq} is the temperature of the surrounding, {eq}T_i {/eq} is the initial temperature of the object and k is the rate constant.


Suppose we have the following:

{eq}\begin{align*} T_i &= H_i^{\circ}\ F &\text{initial temperature of yam}\\ T(t) &= H ^{\circ}\ F &\text{temperature of yam at time, t}\\ T_0 &= 350^{\circ}\ F &\text{temperature of oven}\\ k &= R &\text{ rate constant} \end{align*} {/eq}

We can rewrite our Newton's cooling law as {eq}H^{\circ}\ F = 350^{\circ}\ F + (H_i-350)^{\circ}\ Fe^{-Rt} {/eq}.


We can rearrange the equation to find R as a function of H.


{eq}\displaystyle \begin{align*} H^{\circ}\ F &= 350^{\circ}\ F + (H_i-350)^{\circ}\ Fe^{-Rt} \\ H^{\circ}\ F - 350^{\circ}\ F &= (H_i-350)^{\circ}\ Fe^{-Rt} \\ \frac{ (H - 350 )}{(H_i-350)}&= e^{-Rt} \\ \text{taking the ln of both sides.}\\ \ln \bigg( \frac{ (H - 350 )}{(H_i-350)}&= e^{-Rt} \bigg) \\ \ln \bigg( \frac{ (H - 350 )}{(H_i-350)}\bigg) &= -Rt\\ R &=\boxed{ -\frac{1}{t}\ln \bigg( \frac{ (H - 350 )}{(H_i-350)}\bigg)} \end{align*} {/eq}



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What are Heating and Cooling Curves?

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