What is the possible range for the distance between Hive A and Hive C?
Understand the Problem
The question is asking for the possible range of distances between Hive A and Hive C, given the distances from Hive A to Hive B and from Hive B to Hive C, which likely relates to the triangle inequality theorem.
Answer
The possible range for the distance between Hive A and Hive C is: $488 < x < 1088$.
Answer for screen readers
The possible range for the distance between Hive A and Hive C is: $$ 488 < x < 1088 $$
Steps to Solve
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Identify the known distances The distance between Hive A and Hive B is 788 feet (let's call this $AB$), and the distance between Hive B and Hive C is 300 feet (let's call this $BC$).
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Apply the Triangle Inequality Theorem According to the Triangle Inequality Theorem, for any triangle with sides $a$, $b$, and $c$, the following must hold:
- $a + b > c$
- $a + c > b$
- $b + c > a$
In this context, we are looking for the distance $AC$ (between Hive A and Hive C).
- Set up the inequalities Let the distance between Hive A and Hive C be $x$. We can express the inequalities based on the triangle inequality theorem as follows:
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From $AB + BC > AC$: $$ 788 + 300 > x $$ Simplifying, we get: $$ x < 1088 $$
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From $AB + AC > BC$: $$ 788 + x > 300 $$ Rearranging gives: $$ x > 300 - 788 $$ Since $300 - 788$ is negative, we express this as: $$ x > 0 $$
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From $BC + AC > AB$: $$ 300 + x > 788 $$ Rearranging gives: $$ x > 488 $$
- Combine the inequalities After solving, we can combine our results to get: $$ 488 < x < 1088 $$
The possible range for the distance between Hive A and Hive C is: $$ 488 < x < 1088 $$
More Information
This result tells us that the distance between Hive A and Hive C must be more than 488 feet and less than 1088 feet to satisfy the triangle inequality. This property is essential in geometry, ensuring the three points can form a triangle.
Tips
- Not applying all three inequalities: Some might consider just one or two inequalities from the triangle inequality theorem, which can lead to an incomplete solution.
- Ignoring negative results: When rearranging inequalities, ensure that negative values are handled correctly, as they do not apply in the context of distances.
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