To find a possible distance from Amanda's house (A) to Chad's house (C), we can use the concept of the triangle inequality. In any triangle, the length of one side must be less than the sum of the lengths of the other two sides and greater than the difference of those lengths.
In this situation, we have the following information:
- Distance from A to B (Amanda to the baseball diamond) = 2,200 ft.
- Distance from C to B (Chad to the baseball diamond) = 1,320 ft.
- We need to find the distance from A to C (Amanda to Chad), which we will denote as x.
According to the triangle inequality, we have:
- \( x + 1,320 , ft > 2,200 , ft \) (A + C must be greater than AB)
- \( x + 2,200 , ft > 1,320 , ft \) (A + B must be greater than CB)
- \( 2,200 , ft + 1,320 , ft > x \) (AB + CB must be greater than AC)
Solving each inequality:
-
\( x + 1,320 > 2,200 \)
- \( x > 2,200 - 1,320 \)
- \( x > 880 \)
-
\( x + 2,200 > 1,320 \)
- \( x > 1,320 - 2,200 \)
- \( x > -880 \) (this doesn't impose a useful restriction since x must be positive)
-
\( 2,200 + 1,320 > x \)
- \( 3,520 > x \)
- \( x < 3,520 \)
So from these inequalities, we have:
- \( x > 880 \)
- \( x < 3,520 \)
Thus, the possible distances from A to C (x) can be any value greater than 880 ft and less than 3,520 ft.
Now let's analyze the given options:
- 4,400 ft. - too large, not valid.
- 1,040 ft. - valid (greater than 880 and less than 3,520).
- 880 ft. - not greater than 880, not valid.
- 3,520 ft. - not less than 3,520, not valid.
Based on this, the only possible distance from Amanda's (A) to Chad's (C) is 1,040 ft. Thus, the correct response is 1,040 ft. This distance satisfies the triangle inequality conditions described above.
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