Question

1. The height profile of the landscape somewhere in the Santa Monica Mountains is given by H(x.y) 5x+4xy-2y+18x-24y+99, where H is in meters, and x and y are the coordinates, in kilometers, east | and north from the trailhead, respectively (a) At what location, relative to the trailhead, is the hilltop? (b) How high is this hill? (c) In what direction is the slope steepest at a point which lies 1 km south and 1 km east of the trailhead? What is that steepness (in meters per kilometer)?

Answer 1

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H(d, y) = - 5 x² + 4ny - 2y² +18% - 244 +99 TH= (lox +44 +18) î +(4x -44 - 24) j at the top of hill Th=0 = -lox + 4y +18=0 - 4x-44-2430 = adding both equation we get - 6x-6=0 x = -1 y = -24 -4 --28 hill is located 1km west & 28km south of truil head b) Height of hell H1 -1, -28)= -5 +4 (28)-2/28) + (-18) - 24/ 28 / +99 Ht 28 = 174 m na Ikm y = - 1 km. Dh= (-10 + (-4) +18) î +(4+4 - 24) j = 4î - 169 direction is Southeast ! strepness 17h1 = 16.5 m/km