A hillside trail is represented in the coordinate plane

MATHEMATICS · Subjective · A Level · Mathematics · Coordinate geometry in the (x y) plane · Coordinate geometry in the (x y) plane

Problem

A hillside trail is represented in the coordinate plane. The foot of the hill is at

X(0,0)

and a rest point higher up the hill is at

Y(500,120)

, where the distances are measured in metres. (i) Determine the slope of the segment

XY

and express it both as a fraction and as a percentage grade (correct to one decimal place). (ii) From the rest point

Y

, the trail continues horizontally to a viewpoint at

Z(650,120)

. Find the slope of the segment

YZ

and interpret its value. (iii) Safety guidelines state that pedestrian paths should have an absolute slope less than

0.08

. Decide, with justification, which of the two segments,

XY

YZ

, satisfies this requirement. (iv) Suppose instead that a straight ramp is to be constructed directly from

X

Z

. Calculate the slope of this ramp and hence determine whether it meets the safety guideline stated in part (iii).