$ABCD$ is a rectangular field whose sides, $AB$ and $BC$, measure $2a$ m and $a$ m respectively. A r

Question

$ABCD$ 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]

Accepted Answer

(i) $T=\frac{2a-x}{100}+\frac{\sqrt{a^2+x^2}}{60}\text{ minutes}$ (ii) $x=\frac{3a}{4}$, with $T_{\min}=\frac a{30}\text{ minutes}$

2015 ACJC P1 Q3

2015 ACJC P1 Q3

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