Imagine that you are standing on a hillside somewhere near the equator, and there is a...

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Imagine that you are standing on a hillside somewhere near the equator, and there is a nearby lighthouse. If we use the lighthouse as a landmark, we can set up a useful coordinate system. Let z represent horizontal distance due east from the lighthouse, and y represent horizontal distance due north from the lighthouse, both measured in metres. So the negative x axis points due west. Within 100m of the lighthouse, the altitude of the ground (in metres above sea level) is quite accuately described by the multivariate function. H(x, y) = 200 + 1 250 (-2x + 3xy - y) + sin(Tx-y). 2 You are at the point where x = 40m and y = 25m, with the positive x-axis to the east and the positive y-axis to the north. So you are at height H(40, 25) = 196.7m. (a) Calculate If you head due east, would you initially be ascending or de- scending? At what angle (in degrees) from the horizontal? Show your work. (b) Calculate If you head due south, would you initially be ascending or descending? At what angle (in degrees) from the horizontal? Show your work. (c) Use MATLAB and the surf command to plot H(x, y) on the domain [-100, 100], y [-100, 100]. Label your axes, and include your plot in your answers. (d) Zoom out, and now use MATLAB and the surf command to plot H(x, y) on the domain x [-1000, 1000], ye [-1000, 1000]. List at least two reasons why H(x, y) might not be a good model for a hillside at this larger scale. Imagine that you are standing on a hillside somewhere near the equator, and there is a nearby lighthouse. If we use the lighthouse as a landmark, we can set up a useful coordinate system. Let z represent horizontal distance due east from the lighthouse, and y represent horizontal distance due north from the lighthouse, both measured in metres. So the negative x axis points due west. Within 100m of the lighthouse, the altitude of the ground (in metres above sea level) is quite accuately described by the multivariate function. H(x, y) = 200 + 1 250 (-2x + 3xy - y) + sin(Tx-y). 2 You are at the point where x = 40m and y = 25m, with the positive x-axis to the east and the positive y-axis to the north. So you are at height H(40, 25) = 196.7m. (a) Calculate If you head due east, would you initially be ascending or de- scending? At what angle (in degrees) from the horizontal? Show your work. (b) Calculate If you head due south, would you initially be ascending or descending? At what angle (in degrees) from the horizontal? Show your work. (c) Use MATLAB and the surf command to plot H(x, y) on the domain [-100, 100], y [-100, 100]. Label your axes, and include your plot in your answers. (d) Zoom out, and now use MATLAB and the surf command to plot H(x, y) on the domain x [-1000, 1000], ye [-1000, 1000]. List at least two reasons why H(x, y) might not be a good model for a hillside at this larger scale.