<p>The temperature, \(\theta{}^\circ\)C, of a cup of hot chocolate milk, as it cools to room temperature, can be modelled by the formula \(\theta=27+qb^{-t}\), where&nbsp;\(q\) and \(b\) are constants and \(t\) is the time of cooling in minutes. Measurements of \(t\) and \(\theta\) are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">\(t\)</th><th style="text-align: center; --cell-justify: center">\(2\)</th><th style="text-align: center; --cell-justify: center">\(4\)</th><th style="text-align: center; --cell-justify: center">\(6\)</th><th style="text-align: center; --cell-justify: center">\(8\)</th><th style="text-align: center; --cell-justify: center">\(10\)</th><th style="text-align: center; --cell-justify: center">\(12\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\theta\)</td><td style="text-align: center; --cell-justify: center">\(71\)</td><td style="text-align: center; --cell-justify: center">\(59\)</td><td style="text-align: center; --cell-justify: center">\(48\)</td><td style="text-align: center; --cell-justify: center">\(43\)</td><td style="text-align: center; --cell-justify: center">\(38\)</td><td style="text-align: center; --cell-justify: center">\(35\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain how a straight line graph can be drawn to represent the formula and draw it for the given data.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Estimate the value of \(q\) and of \(b\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Use your graph to estimate the temperature of the cup of tea when&nbsp;\(t=7\)<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">What is the room temperature? Explain how you obtain your answer.<span class="inline-marks">[2]</span></div></li></ol>

The temperature, \(\theta{}^\circ\)C, of a cup of hot chocolate milk, as it cools to room temperature, can be modelled by the formula \(\theta=27+qb^{-t}\), where \(q\) and \(b\) are constants and \(t\) is the time of cooling in minutes. Measurements of \(t\) and \(\theta\) are shown in the table below.

Explain how a straight line graph can be drawn to represent the formula and draw it for the given data.

Estimate the value of \(q\) and of \(b\).

Use your graph to estimate the temperature of the cup of tea when \(t=7\)

What is the room temperature? Explain how you obtain your answer.

Binomial Theorem

Question 9c67c850