Margaret walks to the store using the following path: 0.435 miles west, 0.280 miles north, 0.300 miles east. What is her total displacement? That is, what is the length and direction of the vector that points from her house directly to the store?
Find magnitude in miles and direction in Degrees please???
Thank you
D = (-0.435+0.300) + i0.280.
D = -0.135 + i0.280.
tanAr=Y/X = 0.280 / -0.135=-2.07407,
Ar -64.26 Deg.
A = Ar + 180,
A=-64.26 + 180 = 115.7 Deg.=Direction.
D=X/cosA = -0.135 / cos115.7=0.311 Mi.
@ 115.7 Deg.
To find the total displacement, we need to calculate the magnitude and direction of the vector that points from Margaret's house directly to the store.
First, let's break down Margaret's path into its components along the x-axis (east/west) and y-axis (north/south):
Westward distance: 0.435 miles
Northward distance: 0.280 miles
Eastward distance: 0.300 miles
To find the x-component of the displacement, we subtract the distance travelled to the east from the distance travelled to the west:
x-component = Eastward distance - Westward distance
= 0.300 miles - 0.435 miles
= -0.135 miles (negative because it is westward)
To find the y-component of the displacement, we add the distance travelled to the north and subtract the distance travelled to the south:
y-component = Northward distance
= 0.280 miles
Now, we can calculate the magnitude of the displacement using the Pythagorean theorem:
Magnitude = sqrt((x-component)^2 + (y-component)^2)
= sqrt((-0.135 miles)^2 + (0.280 miles)^2)
= sqrt(0.018225 + 0.0784)
= sqrt(0.096625)
= 0.3106 miles (rounded to four decimal places)
Now, let's calculate the direction of the displacement. We can use the inverse tangent function (arctan) to find the angle relative to the positive x-axis:
Direction = arctan(y-component / x-component)
= arctan(0.280 miles / -0.135 miles)
= arctan(-2.0741)
= -63.96 degrees (rounded to two decimal places)
Note: The negative sign for the direction means that the displacement is west of the positive x-axis.
Therefore, Margaret's total displacement is approximately 0.3106 miles in length and is directed 63.96 degrees west of the positive x-axis.