# CAT Quantitative Aptitude Questions | CAT Time Speed and Distance & Races

###### CAT Questions | Speed Time | Perpendicular directions

The question is from the topic speed time. It also involves ideas from Geometry. CAT exam is known to test on multiple ideas in the same question, so you should be ready to face them. Time Speed and Distance is a favorite in CAT Exam, and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam.

Question 28 : Arjun travels from A to B, a distance of 200 Km at the speed of 40 Km/hr. At the same time, Rakesh starts from point B at a speed of 20 Km/hr along a road which is perpendicular to AB. Find the time in which Arjun & Rakesh will be closer to each other?

1. 1.5 hr
2. 3.33 hr
3. 5 hr
4. 4 hr

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##### Method of solving this CAT Question from Time Speed Distance: Create a visual of where they are and how they travel.

Distance between the two at any time t, D = √((200−40t)2+(20t)2)
If X is minimum, then √will also be minimum and vice versa.
So, we need to minimize (200−40t)2+(20t)2
Or, we need to minimize 400 (20 – 2t)2 + 400t2
Or, we need to minimize (20 – 2t)2 + t2
Or, we need to minimize 400 + 4t2 – 80t + t2
Or, we need to minimize 5t2 – 80t + 400
Or, we need to minimize t2 – 16t + 80
Or, we need to minimize t2 – 16t + 64 + 16
Or, we need to minimize (t – 4)2 + 16
Or, t should be 4.
Or, the distance will be minimum in 4 hours’ time.
WE could do this using differentiation also, but we do not need to know differentiation to answer this question

The question is "Find the time in which Arjun & Rakesh will be closer to each other?"

##### Hence, the answer is 4 hr.

Choice D is the correct answer.

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