A man runs 2.7 km north and then 1.3 km in a direction 24° east of north. A woman walks directly between the same initial and final points.What distance does the woman walk? (km) In what direction does the woman walk? (Enter only positive, acute angles.)
Draw a diagram!
If we label the points
A: start
B: where he turned
C: final location
Then we have a triangle ABC with angle B = 156°
Now just use the law of cosines to find AC.
relative to B, C is at 1.3sin24° East and 1.3cos24° North
So, from A, C is at (x,y) = (2.7+1.3cos24°,1.3sin24°)
So, relative to due East, the angle z is such that
tan z = y/x
To find the distance the woman walks, we can use the Pythagorean theorem to calculate the hypotenuse of the right triangle formed by the man's two displacements.
First, we need to find the components of the man's two displacements - the northward component and the eastward component.
The northward component of the first displacement is obtained by multiplying the distance (2.7 km) by the sine of the angle 24° (measured from the north). So, the northward component of the first displacement is:
Northward component of the first displacement = 2.7 km * sin(24°)
The eastward component of the first displacement is obtained by multiplying the distance (2.7 km) by the cosine of the angle 24° (measured from the north). So, the eastward component of the first displacement is:
Eastward component of the first displacement = 2.7 km * cos(24°)
For the second displacement, the northward component is obtained by multiplying the distance (1.3 km) by the sine of the angle 90° - 24° = 66°. So, the northward component of the second displacement is:
Northward component of the second displacement = 1.3 km * sin(66°)
The eastward component of the second displacement is obtained by multiplying the distance (1.3 km) by the cosine of the angle 90° - 24° = 66°. So, the eastward component of the second displacement is:
Eastward component of the second displacement = 1.3 km * cos(66°)
Now, we have the northward and eastward components of both displacements. We can add these components to find the resulting displacement.
Northward component of the total displacement = Northward component of the first displacement + Northward component of the second displacement
Eastward component of the total displacement = Eastward component of the first displacement + Eastward component of the second displacement
The distance the woman walks is equal to the magnitude of the total displacement, which can be found using the Pythagorean theorem:
Distance the woman walks = sqrt((Northward component of the total displacement)^2 + (Eastward component of the total displacement)^2)
To find the direction the woman walks, we can use trigonometry to calculate the angle between the woman's displacement and the positive x-axis.
Angle = arctan((Northward component of the total displacement) / (Eastward component of the total displacement))
Now, we can substitute the given values into the calculations to find the distance the woman walks and the direction she walks.