​​A robot is designed to move in a peculiar way and it can be set in motion by a microprocessor program. The program can be initiated by assigning a positive rational value to its variable n. The program directs the robot to move in the following way. As soon as the program is started, the robot starts from the point O, moves 2n metres northward and changes its direction by n° to the right. It then moves 2n metres forward and again changes its direction by n° to the right and continues in this manner till it reaches the starting point O, or till it covers a total distance of 1000 m, whichever happens first, and then it stops.​​

a. I assigned a value for n and started the program. If the robot finally came back to O and stopped, what is the total distance that it has covered?
1. 180 m
​2. 360 m
3. 720 m
4. Cannot be determined.

​b. ​​​For how many values of n in the intervals [1, 60] does the robot cover less than 1000 m, before it stops?
1. 19
2. 60
3. 355
​4. Infinte

Dear student,

a)
If n is factor of 360, then according to the pattern of movement followed by the robot,it will cover a regular polygon of an external angle of n° and number of sides=360nThe length of each side will be 2n metres. Hence the robot will come back to O in this case.However, if n is not a factor of 360°, then the robot will not come back to O, but will continue moving till it covers 1000m and then stop.Note: The robot may come back to O for other values of n , which are not factors of 360° but arefactors of 720°, 1080°....etc. However, in such cases the distance required to be covered before reaching O will be greater than 1000m.Since the robot came back to O, n must be a factor of 360° and also total distance covered=no. of sides of the regular polygon×length of each side=360n×2n=720 m
thus option (3) is correct.

b)
If robot cover less than 1000 m, then it must have come back to O. The factorsof 360 in the range [1,60] are 3601=360 sides to 36060=6 sides. All other rational values of n, for 359 sides, 358 sides and so on till 6 sides are possible.Hence a total of 360-6+1=355 values are possible.Thus option (3) is correct.

Regards
 

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