The possible range for the distance between Hive A and Hive C is _ < x < _.

Understand the Problem

The question asks for the possible range of distances between Hive A and Hive C given their distances from Hive B and the constraints of forming a triangle. We will solve it using the triangle inequality theorem.

Answer

$ 150 < x < 1050 $

Steps to Solve

Identify the distances Let ( AB = 600 ) feet (distance from Hive A to Hive B) and ( BC = 450 ) feet (distance from Hive B to Hive C). We want to find the distance ( AC ) (from Hive A to Hive C).
Apply the triangle inequality theorem The triangle inequality theorem states that for any triangle with sides ( a ), ( b ), and ( c ):

( a + b > c )
( a + c > b )
( b + c > a )

In this case, we can set:

( a = AB = 600 ) feet
( b = BC = 450 ) feet
( c = AC ) (we'll denote it as ( x ))

Set up inequalities Using the triangle inequality, we get the following inequalities:

From ( AB + BC > AC ): $$ 600 + 450 > x $$ $$ 1050 > x $$
From ( AB + AC > BC ): $$ 600 + x > 450 $$ $$ x > 450 - 600 $$ $$ x > -150 $$ (which simplifies to just ( x > 0 ), since distance can't be negative)
From ( BC + AC > AB ): $$ 450 + x > 600 $$ $$ x > 600 - 450 $$ $$ x > 150 $$

Combine the inequalities From the inequalities, we see:

( x > 150 )
( x < 1050 )

Thus, the range of possible distances between Hive A and Hive C is: $$ 150 < x < 1050 $$

The possible range for the distance between Hive A and Hive C is ( 150 < x < 1050 ).

More Information

This range reflects the constraints imposed by the triangle inequality theorem, which ensures that the sum of any two sides of a triangle must always be greater than the third side.

Tips

Forgetting the triangle inequality: It's easy to miss the requirement that for three points to form a triangle, distances must adhere to the triangle inequality.
Misinterpreting distances: Ensure you're correctly labeling what each distance represents, as misidentification can lead to incorrect inequalities.

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The possible range for the distance between Hive A and Hive C is ___ < x < ___.