Bunuel wrote:
A certain harbor has docking stations along its west and south docks, as shown in the figure; any two adjacent docking stations are separated by a uniform distance d. A certain boat left the west dock from docking station #2 and moved in a straight line diagonally until it reached the south dock. If the boat was at one time directly east of docking station #4 and directly north of docking station #7, at which docking station on the south dock did the boat arrive?
A. #7
B. #8
C. #9
D. #10
E. #11
Let’s rethink this problem as one that involves the equation of a line. Station 5 is the origin (0,0), and all other stations take on the traditional values on the x and y axes. Thus, when the boat leaves station #2, the ordered pair is (0,3), and the line of the boat’s path passes through the point (station #7, station #4), or (2,1). The line that connects these two points has a slope of (3 – 1)/0 – 2) = 2/-2 = -1. Using y = -1x + b, we substitute the values from (0,3), obtaining 3 = (-1)(0) + b, and so b = 3. Thus, the equation of the line connecting the two points is y = -x + 3.
We now plug in the ordered pairs of the answer choices to determine which of their equivalent ordered pairs satisfies the equation y = -x + 3.
Choice A: station #7 is at (2,0). Does 0 = -2 + 3? No.
Choice B: station #8 is at (3,0). Does 0 = -3 + 3? Yes!
Answer: B
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