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      <text>19.1.1.2 Newton's Law of Cooling</text>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
 <p> Which of the following mathematical equations is equivalent to Newton's Law of Cooling?&nbsp;First, find an expression for the rate of cooling, and an expression for&nbsp;<span style="font-size: 0.9375rem;">the difference between the liquid's temperature and that of the&nbsp;</span>
    <span style="font-size: 0.9375rem;">environment.&nbsp;</span><span style="font-size: 0.9375rem;">Note that \(k\) is a positive constant.</span></p>]]></text>
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      <text><![CDATA[<p>You should obtain <span class="math inline">\({\mathrm{d}\theta\over \mathrm{d}t}\propto (\theta-\theta_\text{s})\)</span> or, equivalently: <span class="math inline">\({\mathrm{d}\theta\over \mathrm{d}t}=-k(\theta-\theta_\text{s})\)</span>.&nbsp;&nbsp;<span class="math inline">\(k\)</span> is a positive constant of proportion and the negative sign is present because (<span class="math inline">\(\theta-\theta_\text{s}\)</span>) is positive, whereas <span class="math inline">\({\mathrm{d}\theta\over \mathrm{d}t}\)</span> must be negative, since <span class="math inline">\(\theta\)</span> decreases with time. The units of <span class="math inline">\(k\)</span> are s<span class="math inline">\(^{-1}\)</span>. The accompanying condition is <span class="math inline">\(\theta=\theta_0\)</span> at <span class="math inline">\(t=0\)</span> which simply states the temperature of the liquid when the cooling begins.</p>]]></text>
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      <text>Your answer is correct.</text>
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    <answer fraction="100" format="html">
      <text><![CDATA[<p dir="ltr" style="text-align: left;">\(\dfrac{\mathrm{d} \theta}{\mathrm{d}t} = -k(\theta - \theta_\text{s})\)</p>]]></text>
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      <text><![CDATA[<p dir="ltr" style="text-align: left;">\(\dfrac{\mathrm{d} \theta}{\mathrm{d}t} = k(\theta - \theta_\text{s})\)<br></p>]]></text>
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      <text><![CDATA[<p dir="ltr" style="text-align: left;">\(\dfrac{\mathrm{d} \theta}{\mathrm{d}t} = k(\theta_\text{s} - \theta)\)<br></p>]]></text>
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      <text><![CDATA[<p dir="ltr" style="text-align: left;">\(\dfrac{\mathrm{d} \theta}{\mathrm{d}t} = -k(\theta_\text{s} - \theta)\)<br></p>]]></text>
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