Jiayi is designing a game that uses a peg board that contains an 11 by 11 rectangular array of peg...
GMAT Two Part Analysis : (TPA) Questions
Jiayi is designing a game that uses a peg board that contains an 11 by 11 rectangular array of peg holes. For design purposes she is modeling the positions of the peg holes on the board as the set of all points \((\mathrm{a},\mathrm{b})\) in the standard \((\mathrm{x},\mathrm{y})\) coordinate plane such that each of a and b is an integer between -5 and 5, inclusive. Set S consists of the rectangular array of those points in the model such that \(-5 \leq \mathrm{a} \leq -2\) and \(-3 \leq \mathrm{b} \leq -2\). Set T consists of the rectangular array of those points in the model such that \(2 \leq \mathrm{a} \leq 3\) and \(-3 \leq \mathrm{b} \leq 4\).
Select for Least distance the least distance between a point in S and a point in T, and select for Greatest distance the greatest distance between a point in S and a point in T. Make only two selections, one in each column.
4
6
\(\sqrt{98}\)
\(\sqrt{17}\)
\(\sqrt{61}\)
\(\sqrt{113}\)
Let me find the least and greatest distances between points in sets S and T.
Set \(S = \{(x,y) : x \in \{-5,-4,-3,-2\}, y \in \{-3,-2\}\}\)
Set \(T = \{(x,y) : x \in \{2,3\}, y \in \{-3,-2,-1,0,1,2,3,4\}\}\)
For the least distance, I need to find the closest points between the two sets. Since the sets are separated horizontally, the minimum distance will occur between points with the same y-coordinate and the closest x-coordinates.
The rightmost x-value in S is -2 and the leftmost x-value in T is 2. Both sets contain y-values -3 and -2.
Closest points: \((-2, -3)\) in S and \((2, -3)\) in T (or \((-2, -2)\) and \((2, -2)\))
Distance = \(\sqrt{[(2-(-2))^2 + (-3-(-3))^2]} = \sqrt{[4^2 + 0^2]} = \sqrt{16} = 4\)
For the greatest distance, I need the farthest points, which will be at opposite corners of the rectangular regions.
Farthest points: \((-5, -3)\) in S and \((3, 4)\) in T
Distance = \(\sqrt{[(3-(-5))^2 + (4-(-3))^2]} = \sqrt{[8^2 + 7^2]} = \sqrt{[64 + 49]} = \sqrt{113}\)
Therefore, the least distance is 4 and the greatest distance is \(\sqrt{113}\).