7.Figure 2 shows a rectangular section of marshland,\(O A B C\) ,which is \(a\) metres long by \(b\) metres wide,where \(a > b\) .
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\includegraphics[alt={},max width=\textwidth]{175528b0-6cd1-4d0d-a6b3-28ac980f74f3-22_360_847_340_609}
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\caption{Figure 2}
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Edgar intends to get from \(O\) to \(B\) in the shortest possible time.In order to do this,he runs along edge \(O A\) for a distance \(x\) metres \(( 0 \leqslant x < a )\) to the point \(D\) before wading through the marsh directly from \(D\) to \(B\) .
Edgar can wade through the marsh at a constant speed of \(1 \mathrm {~ms} ^ { - 1 }\) ,and he can run along the edge of the marsh at a constant speed of \(\lambda \mathrm { ms } ^ { - 1 }\) ,where \(\lambda > 1\)
- By finding an expression in terms of \(x\) for the time taken,\(t\) seconds,for Edgar to reach \(B\) from \(O\) ,show that
$$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { 1 } { \lambda } - \frac { a - x } { \sqrt { ( a - x ) ^ { 2 } + b ^ { 2 } } }$$
- Find,in terms of \(a , b\) and \(\lambda\) ,the value of \(x\) for which \(\frac { \mathrm { d } t } { \mathrm {~d} x } = 0\)
- Show that this value of \(x\) lies in the interval \(0 \leqslant x < a\) provided \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
- For \(\lambda\) in this range,show that the value of \(x\) found in(b)(i)gives a minimum value of \(t\) .
- Find the minimum time taken for Edgar to get from \(O\) to \(B\) if
- \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
- \(1 < \lambda < \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
Edgar's friend,Frankie,also runs at a constant speed of \(\lambda \mathrm { m } \mathrm { s } ^ { - 1 }\) .Frankie runs along the edges \(O A\) and \(A B\) .Given that \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
- find the range of values of \(\lambda\) for which Frankie gets to \(B\) from \(O\) in a shorter time than Edgar's minimum time.