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Get Answers to all your Questions
Q.32) Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of $60 \mathrm{~km} / \mathrm{h}$ in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes fin the opposite direction. Choose the correct-bption for the period T of the bus service and the speed (assumed constant) of the buses.
A) $15 \mathrm{~min}, 120 \mathrm{~km} / \mathrm{h}$
B) $9 \mathrm{~min}, 40 \mathrm{~km} / \mathrm{h}$
C) $25 \mathrm{~min}, 100 \mathrm{~km} / \mathrm{h}$
D) $10 \mathrm{~min}, 90 \mathrm{~km} / \mathrm{h}$
Answers (1)
In same direction, relative speed $=v-60$, time between buses $=30 \mathrm{~min} \rightarrow$ distance between buses $=(v-$ $60) \times 0.5$
So:
$$
\text { Distance between buses }=v \cdot \frac{T}{60}=(v-60) \cdot 0.5
$$
In opposite direction, relative speed $=v+60$, time $=10 \mathrm{~min} \rightarrow$ distance $=(v+60) \cdot \frac{1}{6}$
So:
$$
v \cdot \frac{T}{60}=(v+60) \cdot \frac{1}{6}
$$
From (1):
$$
v \cdot \frac{T}{60}=\frac{v-60}{2} \Rightarrow \frac{v T}{60}=\frac{v-60}{2}
$$
From (2):
$$
\frac{v T}{60}=\frac{v+60}{6}
$$
Equating (a) and (b):
$$
\frac{v-60}{2}=\frac{v+60}{6} \Rightarrow 3(v-60)=v+60 \Rightarrow 3 v-180=v+60 \Rightarrow 2 v=240 \Rightarrow v=120 \mathrm{~km} / \mathrm{h}
$$
Plug into (a):
$$
\frac{120 T}{60}=\frac{120-60}{2} \Rightarrow 2 T=30 \Rightarrow T=15 \mathrm{~min}
$$
Correct answer: (1) $15 \mathrm{~min}, 120 \mathrm{~km} / \mathrm{h}$
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