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Apersondesirestoreacha pointthatis3.42kmfromher presentlocationand in a directionthat is 35.0°northof east However,shemusttravelalongstreetsthatgoeithernorth- southoreast-west.What is the minimumdistanceshecould travel to reachher destination?

Apersondesirestoreacha pointthatis3.42kmfromher presentlocationand in a directionthat is 35.0°northof east
However,shemusttravelalongstreetsthatgoeithernorth- southoreast-west.What is the minimumdistanceshecould travel to reachher destination?

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
6 years ago
Let us assume that the destination of the person relative to its present position be represented by position vector\overrightarrow{a} . We also assume that the magnitude of the vector is represented by a , the horizontal component of the vector be represented by ax whereas the vertical component of the vector be represented by ay .
We also assume that the angle subtended by the vector\overrightarrow{a} with respect to the positive x axis, along the unite vector \widehat{i}is represented by \o.
Given:
a = 3.42 km
\o = 35.0^{\circ}
The figure below shows the vector\overrightarrow{a} , directed at 35.0° north of east. The components of the vectors are also shown in the figure. The horizontal component of vector \overrightarrow{a}is given as a\ cos\ \owhereas the vertical component (ay)of the vector \overrightarrow{a}is given asa\ sin\ \o

236-1541_1.PNG

It is important to note that the magnitude of the component of vector\overrightarrow{a} represents the shortest distance that the person should travel to reach her destination in minimum time, However, with the constraints of moving either north-south or east-west, the minimum distance will be equal to the sum of horizontal component of vector and the vertical component of vector
The horizontal vector component of vector \overrightarrow{a} is:
a_{x}\widehat{i} = a\ cos\ \o \widehat{i}
Substituting the given values, we have
a_{x}\widehat{i} = (3.42\ km)\ cos\ (35.0^{\circ})\widehat{i}
= 2.80\ km\ \widehat{i}
The vertical vector component of vector \overrightarrow{a}is:
a_{y}\widehat{j} = a\ sin\ \o\ \widehat{j}
Substituting the given values, we have
a_{y}\widehat{j} = (3.42\ km)sin(35.0^{\circ})
= 1.96 km\ \widehat{j}
Therefore the person must walk 1.96 km east and 2.80 km north to reach her destination.
The total distance (say s) traveled by the person is:
s = ax + ay
= 2.80 km + 1.96km
The person can choose to either walk north first and then east, however this does not affect the distance she travels.
Therefore the total minimum distance traveled by the person is 4.76km.

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