A person walks 4.0 miles east and then makes a 50° turn to the right and walks another 2.0 miles. How far is she from her starting point?

east distance = x = 4 + 2 cos 50

south distance = y = -2 sin 50

d = sqrt ( x^2 + y^2 )

To find out how far the person is from her starting point, we can use the concept of vector addition and trigonometry.

First, let's break down the distances and directions of the person's movements:

1. The person walks 4.0 miles east. We can represent this as a vector (a quantity with magnitude and direction) of 4.0 miles in the positive x-axis direction, which we'll call vector A.

2. Then, the person makes a 50° turn to the right and walks another 2.0 miles. We can represent this as a vector of 2.0 miles in a new direction, which we'll call vector B.

To determine how far the person is from her starting point, we need to find the resultant vector or the sum of vector A and vector B.

Using trigonometry, we can find the vector components of A and B:

- The x-component of vector A is 4.0 miles since it's in the positive x-axis direction.
- The y-component of vector A is 0 miles since it's in the x-axis direction.

To find the components of vector B, we can use the angle of 50°:

- The x-component of vector B is 2.0 miles cos(50°) since it uses the adjacent side of the angle.
- The y-component of vector B is 2.0 miles sin(50°) since it uses the opposite side of the angle.

Next, we can sum the x-components and y-components separately:

- The x-component of the resultant vector is the sum of the x-components of A and B.
- The y-component of the resultant vector is the sum of the y-components of A and B.

Finally, we can use the Pythagorean theorem to find the magnitude (distance) of the resultant vector:

Magnitude = sqrt(x^2 + y^2)

By performing these calculations, we can determine the distance the person is from her starting point.