Question

13) A robot stands on a large paved area. Beneath its feet is line that runs 39.96621 norh of east. As per its instructions it moves 11.29988° west of north for 10m. It then moved 4.61685° south of east until it reached the line. It then stopped. How far was the robot from its original position? (Hint: Tilt the coordinate system.)

Answer 1

The diagram for the given problem is shown below (making origin as initial position of the robot):

The diagram for the given problem is shown below (making origin as initial position of the robot): Now let's tilt the diagram by 11.299880�in clockwise direction. We will get this new diagram: We have to calculate the value of OB . Here, OA = 10 m. First we will derive equation of the line AB: The coordinate of point A is (0, 10). The slope of the line AB can be calculated as m_AB = tan(360 degree - 15.91673 degree) = tan(344.08327 degree) = -0.2851732 so, the equation of line AB can be written as y - 10 = -0.2851732(x - 0) or, y = 10 - 0.2851732 x ----(1) Similarly, we can write equation of line OB. The slope of OB will be m_)B = tan(28.66633 degree) = 0.54672046 As the line OB passes through the origin (0, 0), we can write the equation of the line OB as y - 0 = 0.54672046 (x - 0) or, y

Now let's tilt the diagram by 11.29988⁰ in clockwise direction. We will get this new diagram :

We have to calculate the value of OB . Here, OA = 10 m.

First we will derive equation of the line AB :

The coordinate of point A is (0 , 10).

The slope of the line AB can be calculated as

  

So, the equation of line AB can be written as

or, . ....................................(1)

Similarly, we can write equation of line OB.

The slope of OB will be

  

As the line OB passes through the origin (0 , 0), we can write the equation of the line OB as

or, ........................................(2)

Now, by solving equation (1) and (2), we can calculate the point of intersection of the line OB and AB or the coordinate of point B .

Putting the value of y from equation (2) in equation (1), wee get

or,  

or,

Putting the value of x in equation (2), we get

So, the coordinate of point B is (12.020767 , 6.571999) .

Now, we can calculate the length of the line OB as

  

Hence, the robot is 13.70 m far away from its original position.

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