Margaret walks to the store using the following path: 0.440 miles west, 0.245 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?
To find Margaret's total displacement, we need to find the length and direction of the vector that points from her house directly to the store.
First, let's break down Margaret's path into horizontal (west-east) and vertical (north-south) components.
Horizontal component: Margaret walks 0.440 miles west and then 0.300 miles east. So, the net horizontal displacement is 0.440 - 0.300 = 0.140 miles west.
Vertical component: Margaret walks 0.245 miles north.
Now, we can use these components to find the length and direction of the total displacement vector.
The length of the total displacement vector is given by using the Pythagorean theorem, which states that the square of the length of the hypotenuse (in this case, the total displacement) is equal to the sum of the squares of the other two sides (horizontal and vertical components).
Length = √(horizontal^2 + vertical^2)
= √(0.140^2 + 0.245^2)
≈ √(0.0196 + 0.060025)
≈ √0.079625
≈ 0.282 miles (rounded to three decimal places)
The direction of the total displacement vector can be found using trigonometry. We can use the tangent function to find the angle between the total displacement vector and the positive x-axis (east).
Direction = arctan(vertical / horizontal)
= arctan(0.245 / 0.140)
≈ arctan(1.75)
≈ 59.04 degrees (rounded to two decimal places)
Therefore, Margaret's total displacement is approximately 0.282 miles in a direction of 59.04 degrees (measured counterclockwise from the positive x-axis, east).