<p>The diagram shows a lake (shaded) bordered by a straight level road \(AXB\).</p>
<p>Point \(C\) is on the surface of the lake and \(B\) is the point closest to \(C\).</p>
<p>The distance \(AB\) is \(30\) m and \(BC\) is \(20\) m.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1772946161270-i42w3h2gof.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Natalie is at point \(A\) on the road and wants to get to point \(C\) as quickly as possible. She can run at a speed of \(4\) m/s and swim at a speed of \(2\) m/s.</p>
<p>She realizes that in order to minimize the total time to get to \(C\), it is necessary to leave the road at some point \(X\) between \(A\) and \(B\), and then swim directly to \(C\).</p>
<p>Let&nbsp;\(AX\) be \(x\) m and \(T\) be the total time, in seconds, taken by Natalie to get from \(A\) to \(X\) and then from \(X\) to \(C\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(T = \frac{x}{4} + \frac{1}{2} \sqrt{x^2 - 60x + 1300}\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find an expression for&nbsp;\(\frac{dT}{dx}\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the value of \(x\) for which the total time taken is minimum. You are <strong>not</strong> required to justify that this value of \(x\) leads to the minimum total time.<span class="inline-marks">[4]</span></div></li></ol>

The diagram shows a lake (shaded) bordered by a straight level road \(AXB\).

Point \(C\) is on the surface of the lake and \(B\) is the point closest to \(C\).

The distance \(AB\) is \(30\) m and \(BC\) is \(20\) m.

Natalie is at point \(A\) on the road and wants to get to point \(C\) as quickly as possible. She can run at a speed of \(4\) m/s and swim at a speed of \(2\) m/s.

She realizes that in order to minimize the total time to get to \(C\), it is necessary to leave the road at some point \(X\) between \(A\) and \(B\), and then swim directly to \(C\).

Let \(AX\) be \(x\) m and \(T\) be the total time, in seconds, taken by Natalie to get from \(A\) to \(X\) and then from \(X\) to \(C\).

Show that \(T = \frac{x}{4} + \frac{1}{2} \sqrt{x^2 - 60x + 1300}\).

Find an expression for \(\frac{dT}{dx}\).

Find the value of \(x\) for which the total time taken is minimum. You are **not** required to justify that this value of \(x\) leads to the minimum total time.

2025 DSS PRELIMS P2 Q11

2025 DSS PRELIMS P2 Q11