Suppose you are climbing a hill whose shape is given by the...

Suppose you are climbing a hill whose shape is given by the equation z = 1400 - 0.005x^2 - 0.01y^2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1264). The positive x-axis points east and the positive y-axis points north. (a) If you walk due south, will you start to ascend or descend? ascend descend At what rate? vertical meters per horizontal meter (b) If you walk northwest, will you start to ascend or descend? ascend descend At what rate? (Round your answer to two decimal places.) vertical meters per horizontal meter (c) In which direction is the slope largest? What is the rate of ascent in that direction? vertical meters per horizontal meter At what angle above the horizontal does the path in that direction begin? (Round your answer to two decimal places.)

Transcript

00:05 Now here in this question, suppose you're climbing here, right? her shape is given by the equation, showing in here, all right? and you are currently standing at a point with coordinates of this.

00:17 So the positive x -axis points to the east.

00:21 So this would be east, right? that's the x -axis, and the y points to the north, right? of north, and you stand somewhere, obviously, somewhere here, right? that's where you stand now.

00:33 And you work due source, that means you're going, if you walk it downward like this, right, where you start to ascend or descend, right? so basically what you need to do is to work out the partial derivative of z with respect to y only way, because you are working due source, right? so if you do that, you will find this to be given minus 0 .0 to you are going to do.

00:57 You are going to go sauce, that means why is going to decrease, right? why is going to decrease, actually, this is going to increase.

01:03 So actually you're going to ascend, right? and the second, at what rate, the rate of course is given by this, 0 .02, right? 0 .02, vertical meters per horizontal meter.

01:17 If you work in northwest, now if you work in northwest, north -west, that's going this direction, right? and you work in that direction where you would start to ascend or to some, well, in that case, you really have to work out the gradient of partiality of z, right? so the gradient has two, has x component, the y component, the x point is given by partial t over partial x, and the y company is going partial t over partial y, right? and you work that out is obviously given by minus 0 .01x and a y component, and we already work it out in minus 0 .0 .02, sorry, 02, right? and then if you walk to the north -west, that means your change in your coordinates, let's say, is i call it delta, right? and for the x direction actually is negative, right? so it's minus delta.

02:11 For the wire direction is data, right? so the total change will be actually given by the dot product of this gradient and this vector.

02:18 So that's given by, you know, multiply this with this, you get 0 .01 data.

02:23 And multiply this two, you get minus 0 .02 data...

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