Question

The public transportation department measures the distance from a location using displacement vectors. To ride the bus to work for a discounted amount, you have to travel further than 1.5 mi. (7,920 ft. ) from your home. Your cell phone calculates that when you walk to work, you go 1,600 ft. south and 6,300 ft. west. What is the magnitude of the displacement vector from your home to work, and will you be able to ride the bus for the discounted rate?
(1 point)
Responses

6,500 ft., no discount
6,500 ft., no discount

7,900 ft., no discount
7,900 ft., no discount

42,250 ft., discounted rate
42,250 ft., discounted rate

65,000 ft., discounted rate

Answer 1

To find the magnitude of the displacement vector from your home to work based on the distances traveled south and west, you can use the Pythagorean theorem. The displacement vector can be calculated as follows:

1. **Identify the components of the vector**:
- South: 1,600 ft (this will be along the y-axis).
- West: 6,300 ft (this will be along the x-axis).

2. **Apply the Pythagorean theorem**:
\[
d = \sqrt{(x^2 + y^2)}
\]
where \( x = 6,300 \) ft and \( y = 1,600 \) ft.

3. **Perform the calculation**:
\[
d = \sqrt{(6300^2 + 1600^2)}
\]
\[
= \sqrt{(39690000 + 2560000)}
\]
\[
= \sqrt{42250000}
\]
\[
= 6500 \text{ ft}
\]

Now, we compare the magnitude of the displacement vector to the threshold for the discounted rate:

- The calculated distance is 6,500 ft.
- The requirement for a discounted rate is to travel further than 7,920 ft.

Since 6,500 ft. is less than 7,920 ft., you will **not** be able to ride the bus for the discounted rate.

Thus, the answer is:
**6,500 ft., no discount**.