To determine a possible distance from Amanda's (A) to Chad's (C) based on the triangle inequality theorem, we need to recognize that for any triangle with sides a, b, and c:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
In the context of this problem, let:
- Side \( AC = c \) (distance from Amanda to Chad)
- Side \( AB = b = 2,200 \) ft (distance from Amanda to the baseball diamond)
- Side \( BC = a = 1,320 \) ft (distance from Chad to the baseball diamond)
We can use the triangle inequalities to solve for \( AC \) (the distance between Amanda and Chad) as follows:
Apply the triangle inequalities:
-
From \( a + b > c \):
- \( 1,320 + 2,200 > AC \)
- \( 3,520 > AC \)
- So, \( AC < 3,520 \) ft.
-
From \( a + c > b \):
- \( 1,320 + AC > 2,200 \)
- \( AC > 2,200 - 1,320 \)
- \( AC > 880 \) ft.
-
From \( b + c > a \):
- \( 2,200 + AC > 1,320 \)
- This inequality will always be true for positive distances since \( 2,200 \) ft is already greater than \( 1,320 \) ft.
Summary of findings:
From the inequalities, we found:
- \( AC > 880 \) ft.
- \( AC < 3,520 \) ft.
Evaluate the options:
Now we can evaluate the options:
- 4,400 ft: Too large (not valid).
- 1,040 ft: Valid (between 880 ft and 3,520 ft).
- 880 ft: Not valid (it must be greater than 880 ft).
- 3,520 ft: Not valid (it must be less than 3,520 ft).
Conclusion:
The only valid option between Amanda and Chad is 1,040 ft. Hence, the correct answer is 1,040 ft.
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