<p>The rate of decrease of atmospheric pressure \(P\) Pascals (Pa) with respect to altitude \(h\) kilometres above sea level, is proportional to \(\frac{P}{T}\), where \(T\) is the temperature measured in Kelvin (K). The temperature \(T\) decreases linearly by \(6.5\) K for every one-kilometre increase in the altitude above sea level. It is given that the temperature is \(293\) K at sea level.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a differential equation for \(\frac{\mathrm{d}P}{\mathrm{d}h}\) in terms of \(P\) and \(h\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that \(P&gt;0\), solve this model in part (i) to show that \(P=A|293-6.5h|^b,\) where \(A\) and \(b\) are constants. Explain whether \(b\) is positive or negative.<span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that the atmospheric pressures are \(101300\) Pa at sea level and \(80000\) Pa at \(h=2\).</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the atmospheric pressure at the top of Mount Everest with an altitude of \(8848\) metres.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">For \(h\ge 0\), sketch the graph of \(P\) against \(h\). Explain a limitation of this model.</span><span class="inline-marks">[3]</span></div></li></ol>

The rate of decrease of atmospheric pressure \(P\) Pascals (Pa) with respect to altitude \(h\) kilometres above sea level, is proportional to \(\frac{P}{T}\), where \(T\) is the temperature measured in Kelvin (K). The temperature \(T\) decreases linearly by \(6.5\) K for every one-kilometre increase in the altitude above sea level. It is given that the temperature is \(293\) K at sea level.

Find a differential equation for \(\frac{\mathrm{d}P}{\mathrm{d}h}\) in terms of \(P\) and \(h\).

Given that \(P>0\), solve this model in part (i) to show that \\[P=A|293-6.5h|^b,\\] where \(A\) and \(b\) are constants. Explain whether \(b\) is positive or negative.

Given that \(P>0\), solve this model in part (i) to show that \(P=A|293-6.5h|^b,\) where \(A\) and \(b\) are constants. Explain whether \(b\) is positive or negative.

It is given that the atmospheric pressures are \(101300\) Pa at sea level and \(80000\) Pa at \(h=2\).

Find the atmospheric pressure at the top of Mount Everest with an altitude of \(8848\) metres.

For \(h\ge 0\), sketch the graph of \(P\) against \(h\). Explain a limitation of this model.

2025 NJC P2 Q4

2025 NJC P2 Q4