Respuesta :
Answer:North-East
Step-by-step explanation:
First she walk 3 miles west and then 2 miles southwest
i.e. she is now Southwest direction with respect to initial position
she vertically 2miles below initial position
and horizontally 3+2cos45 away from initial position
[tex]tan\theta =\frac{2}{3+2cos45} [/tex]
Where \theta is angle made by her with horizontal
[tex]\theta =24.37 [/tex]
she should walk at [tex] \theta =24.37[/tex] North of east to reach directly home
The distance that women need to cover is 4.635 miles, while the angle between her current position and her initial position is 17.76°.
What is Tangent (Tanθ)?
The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. it is given as,
[tex]\rm Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]
where,
θ is the angle,
Perpendicular is the side of the triangle opposite to the angle θ,
The base is the adjacent smaller side of the angle θ.
The length of the BD can be written as,
[tex]\rm Sin(\theta) = \dfrac{Perpendicular}{Hypotenuse}\\\\Sin(\angle CBD) = \dfrac{CD}{CB}\\\\Sin(45^o) = \dfrac{CD}{2}\\\\CD = 1.414\ miles[/tex]
Now, the length of the CE is 4.414 miles (3+1.414). Therefore, the horizontal distance of women from the initial point is 4.414 miles.
Further, the length of AE can be written as,
[tex]\rm Cos(\theta) = \dfrac{Base}{Hypotenuse}\\\\Sin(\angle CBD) = \dfrac{BD}{CB}\\\\Cos(45^o) = \dfrac{BD}{2}\\\\BD= 1.414\ miles[/tex]
Now, in the ΔCAE, the length of side AC can be written as,
[tex]\rm AC^2= AE^2+EC^2\\\\AC^2= BD^2+(ED+DC)^2\\\\AC^2=1.414^2+4.414^2\\\\AC^2 = 21.483\\\\AC = 4.635\ miles[/tex]
The angle between the direction of the woman when she is moving southwest and the initial position of the woman can be written as,
[tex]\rm Tan(\theta)= \dfrac{Perpendicular}{Base}\\\\Tan(\angle ACE)= \dfrac{AE}{EC}\\\\(\angle ACE)= Tan^{-1}\dfrac{1.414}{4.414}\\\\(\angle ACE)=17.76^o[/tex]
Thus, the distance that women need to cover is 4.635 miles, while the angle between her current position and her initial position is 17.76°.
Learn more about Tangent (Tanθ):
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