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Sample Question 11:
Step 1: Define the starting position of both bees as (0,0,0) in a three-dimensional coordinate system.
Step 2: Run the simulation. At the end of 6 steps, bee \(A\) is at (2,2,2), having repeated its route twice, while bee \(B\) is at (-3,-3,0), having repeated its route three times. The distance between them is then \(\sqrt{(2+3)^2+(2+3)^2+2^2}=\sqrt{54} \), which is less than 10 feet.
Step 3: Move both bees one step further. In step 7, bee \(A\) moves north, ending up at (2,3,2) and bee \(B\) moves south, ending up at (-3,-4,0). Compute their distance \(\sqrt{(2+3)^2+(3+4)^2+2^2}=\sqrt{78}\), which is still less than 10 feet.
Step 4: Move both bees another step. In step 8, bee \(A\) moves east, ending up at (3,3,2) and bee \(B\) moves west, ending up at (-4,-4,0). Now the distance between them becomes \(\sqrt{(3+4)^2+(3+4)^2+2^2}=\sqrt{102}\), which is greater than 10 feet.
Step 5: This means that the distance between the bees first exceeds 10 feet between the 7th and 8th steps. At that point, bee \(A\) is moving east, and bee \(B\) is moving west. Therefore, answer choice \(\textbf{(A)}\) is correct.
Two bees start at the same spot and fly at the same rate in the following directions. Bee \(A\) travels \(1\) foot north, then \(1\) foot east, then \(1\) foot upwards, and then continues to repeat this pattern. Bee \(B\) travels \(1\) foot south, then \(1\) foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly \(10\) feet away from each other?
\(\textbf{(A)}\ A\) east, \(B\) west
\(\textbf{(B)}\ A\) north, \(B\) south
\(\textbf{(C)}\ A\) north, \(B\) west
\(\textbf{(D)}\ A\) up, \(B\) south
\(\textbf{(E)}\ A\) up, \(B\) west
Answer Keys
Question 11: ASolutions
Question 11Step 1: Define the starting position of both bees as (0,0,0) in a three-dimensional coordinate system.
Step 2: Run the simulation. At the end of 6 steps, bee \(A\) is at (2,2,2), having repeated its route twice, while bee \(B\) is at (-3,-3,0), having repeated its route three times. The distance between them is then \(\sqrt{(2+3)^2+(2+3)^2+2^2}=\sqrt{54} \), which is less than 10 feet.
Step 3: Move both bees one step further. In step 7, bee \(A\) moves north, ending up at (2,3,2) and bee \(B\) moves south, ending up at (-3,-4,0). Compute their distance \(\sqrt{(2+3)^2+(3+4)^2+2^2}=\sqrt{78}\), which is still less than 10 feet.
Step 4: Move both bees another step. In step 8, bee \(A\) moves east, ending up at (3,3,2) and bee \(B\) moves west, ending up at (-4,-4,0). Now the distance between them becomes \(\sqrt{(3+4)^2+(3+4)^2+2^2}=\sqrt{102}\), which is greater than 10 feet.
Step 5: This means that the distance between the bees first exceeds 10 feet between the 7th and 8th steps. At that point, bee \(A\) is moving east, and bee \(B\) is moving west. Therefore, answer choice \(\textbf{(A)}\) is correct.