Question

Point $A$ lies somewhere within or on the square which has opposite corners at $(0,0)$ and $(2,2)$. Point $B$ lies somewhere within or on the square which has opposite corners at points $(4,2)$ and $(5,3)$. What is the greatest possible value of the slope of the line containing points $A$ and $B$? Express your answer as a common fraction.

Answer 1

Maximize m = (y_B - y_A)/(x_B - x_A) with (x_A,y_A) in [0,2]×[0,2] and (x_B,y_B) in [4,5]×[2,3]. The numerator ≤ 3 (take y_B=3, y_A=0) and the denominator ≥ 2 (take x_B=4, x_A=2), so m ≤ 3/2. This is attained at A=(2,0), B=(4,3). Thus the greatest possible slope is 3/2.