To determine the magnitude of the car's displacement after it drives 16 miles south and then 12 miles west, we need to use the Pythagorean theorem. This theorem is used to find the length of the hypotenuse of a right triangle when the lengths of the other two sides are known.
1. Identify the sides of the triangle:
- The car's southward journey of 16 miles can be considered one leg of the right triangle.
- The car's westward journey of 12 miles can be considered the other leg of the right triangle.
2. Apply the Pythagorean theorem:
- The Pythagorean theorem states: [tex]\(a^2 + b^2 = c^2\)[/tex]
- Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the legs of the triangle, and [tex]\(c\)[/tex] is the hypotenuse (the displacement).
3. Plug the numbers into the formula:
- Let [tex]\(a = 16\)[/tex] miles and [tex]\(b = 12\)[/tex] miles.
- Substitute the values into the equation: [tex]\(16^2 + 12^2 = c^2\)[/tex].
4. Perform the calculations:
- Calculate [tex]\(16^2\)[/tex]: [tex]\(16^2 = 256\)[/tex]
- Calculate [tex]\(12^2\)[/tex]: [tex]\(12^2 = 144\)[/tex]
- Add these results: [tex]\(256 + 144 = 400\)[/tex]
5. Find [tex]\(c\)[/tex]:
- [tex]\(c^2 = 400\)[/tex]
- To find [tex]\(c\)[/tex], take the square root of 400: [tex]\(\sqrt{400} = 20\)[/tex]
So, the magnitude of the car's displacement is 20 miles.
The correct answer is:
- O 20 miles
