Sarah walks 5 miles 35 degrees north of east, and then decides to walk 6 miles due west. What is her final displacement?
I drew it out, used the Pythagorean theorem and ended up with square root -11 which is obviously wrong. I'm really confused as to what to do with the angle too. I'm not sure which next steps to take.
I think it might be 5. I don't know.
To find Sarah's final displacement, we need to use vector addition. Let's break down the problem step by step:
1. Convert the given angle to a vector notation. Sarah walks 5 miles at an angle of 35 degrees north of east. To convert this into vector form, we can break it down into its horizontal and vertical components.
The horizontal component is 5 miles multiplied by the cosine of 35 degrees. So the horizontal component is 5 * cos(35°).
The vertical component is 5 miles multiplied by the sine of 35 degrees. So the vertical component is 5 * sin(35°).
2. Calculate the horizontal and vertical components using a calculator or trigonometric tables.
3. Now, we have two vectors:
- The first vector is the displacement from Sarah's starting point to the point 5 miles 35 degrees north of east.
- The second vector is the displacement of 6 miles due west.
4. Represent each vector as a coordinate pair (x, y), where x represents the horizontal component and y represents the vertical component.
For the first vector, the coordinate pair would be (5 * cos(35°), 5 * sin(35°)).
For the second vector, the coordinate pair would be (-6, 0) since she is walking due west.
5. Add the coordinates of the two vectors together.
To do this, add the x-components and the y-components separately.
(x1 + x2, y1 + y2) = ((5 * cos(35°)) + (-6), (5 * sin(35°)) + 0).
6. Simplify the equation.
Final x-component: (5 * cos(35°)) + (-6)
Final y-component: (5 * sin(35°)) + 0
7. Use the Pythagorean theorem to find the magnitude of the final displacement. The magnitude (or length) of the final displacement vector is given by the equation:
Magnitude = sqrt((Final x-component)^2 + (Final y-component)^2).
Plug in the values of the final x-component and final y-component into the formula.
Magnitude = sqrt(((5 * cos(35°)) + (-6))^2 + ((5 * sin(35°)) + 0)^2).
Calculate this expression using a calculator.
The magnitude of the final displacement should give you a positive and realistic answer, indicating the distance and direction from the starting point to the final destination of Sarah's walk.