$ABCD$ is a rectangular field whose sides, $AB$ and $BC$, measure $2a$ m and $a$ m respectively. A road runs along the side $AB$. A man, starting from $A$, wishes to reach the opposite corner $C$ in the shortest possible time. He can walk along the road at 100 m per minute and across the field at 60 m per minute. Find an expression for the time, in minutes, he will take if he walks along the road to $P$, a point $x$ m from $B$, and then across the field from $P$ to $C$.[2]
Use differentiation to find, in terms of $a$, the value of $x$ for the time taken to be the shortest possible. Find, also, the shortest possible time taken, and prove that it is the minimum.[4]

$ABCD$ is a rectangular field whose sides, $AB$ and $BC$, measure $2a$ m and $a$ m respectively. A road runs along the side $AB$. A man, starting from $A$, wishes to reach the opposite corner $C$ in the shortest possible time. He can walk along the road at 100 m per minute and across the field at 60 m per minute. Find an expression for the time, in minutes, he will take if he walks along the road to $P$, a point $x$ m from $B$, and then across the field from $P$ to $C$.

Use differentiation to find, in terms of $a$, the value of $x$ for the time taken to be the shortest possible. Find, also, the shortest possible time taken, and prove that it is the minimum.

2015 ACJC P1 Q3

2015 ACJC P1 Q3

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