



Tools  Math 'Creating Measures' Sharpness Task, Example #6 (solution)
Squareness, Example #1 (solution)  Steepness, Example #2 (solution) Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution) Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
Malcolm Swan
Jim Ridgway
Initially it makes sense to estimate the difficulty of a bend by the angle it turns through ( ). Greater angles give greater difficulty:
At this stage, most students will probably try to answer the question by eye. However, (anticipating question 4), if you try to draw the largest circular arc through each bend, you find that bends A and B will both allow arcs of radius 3.4 cm. Bend C will permit an arc of 6.8 (double the radius for bends A and B) and bend D will permit an arc of radius 13.1 cm.
Thus, the order of difficulty of the bends is now:
It may be seen that one possible measure for the 'difficulty' of a bend is radius of the largest circular arc that may be inscribed within the bend, that is tangential to the outer edges of the road. The only drawback is that the larger this radius, r, the easier the bend. A measure which takes larger values for more difficult bends would be ^{1}/_{r}. This would take the'difficulty' value 0 for a straight road. The most difficult turn would be a U turn (when = 180^{o}). In this case r Alternatively, one could define a measure using two variables, by reasoning in the following way:
Thus we could define difficulty by a proportional model such as:
See answer to question 2.
Suppose the road is of width w, and turns through an angle (which lies between 0^{o} and 180^{o}). We need to find the radius of the circular arc, r (r > w) which is tangential to the outside edges of the road and which just passes through the corner on the inside of the road (Q) in terms of w and .
OP = r  w OQ = r < POQ = ^{}/_{2} (since OPQ is congruent to ORQ)
Notice that r is directly proportional to w. (This may also be seen, for example, by considering road C to be an enlargement of road A by scale factor 2, thus enlarging the radius by the same scale factor). As increases from 0 to 180^{o}, cos (^{}/_{2}) decreases from 1 to 0, so r decreases from an infinite radius to a radius of w.This corresponds to a U turn in a straight road width 2w.
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution) Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)

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