While following the directions on a treasure map, a pirate walks 45.0 m north and then turns and walks 7.5 m east. What single straight-line displacement could the pirate have taken to reach the treasure?

To determine the single straight-line displacement that the pirate could have taken to reach the treasure, we can use the Pythagorean theorem. The pirate's movement can be represented as a right triangle where the legs are the distances walked north and east.

1. The distance walked north (vertical leg) = 45.0 m
2. The distance walked east (horizontal leg) = 7.5 m

The displacement is the hypotenuse of the right triangle formed by these movements.

Using the Pythagorean theorem:

\[
\text{Displacement} = \sqrt{(45.0 \, \text{m})^2 + (7.5 \, \text{m})^2}
\]

First, we calculate the squares of the legs:

\[
(45.0 \, \text{m})^2 = 2025.0 \, \text{m}^2
\]

\[
(7.5 \, \text{m})^2 = 56.25 \, \text{m}^2
\]

Now, we add these values:

\[
2025.0 \, \text{m}^2 + 56.25 \, \text{m}^2 = 2081.25 \, \text{m}^2
\]

Finally, we take the square root of the sum to find the displacement:

\[
\text{Displacement} = \sqrt{2081.25 \, \text{m}^2} \approx 45.6 \, \text{m}
\]

So, the single straight-line displacement the pirate could have taken to reach the treasure is approximately 45.6 meters.

Do you want any further details or have any questions?

Here are some related questions you might want to ask:
1. How do you find the angle of displacement in such problems?
2. What is the Pythagorean theorem, and how is it applied in different contexts?
3. How can trigonometry be used to solve similar displacement problems?
4. What other methods can be used to calculate straight-line displacement?
5. Can you provide more examples of solving displacement problems?

**Tip:** When dealing with displacement problems, always visualize the movements as vectors on a coordinate system; this helps in applying the Pythagorean theorem or trigonometric functions accurately.

Learn how to calculate the straight-line displacement of a pirate using the Pythagorean theorem. This problem involves determining the shortest distance from a series of movements, including 45.0 m north and 7.5 m east. Suitable for students in Grades 6-8.

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