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An ant at the bottom of an almost empty sugar bowl eats the last few remaining grains. It is now too bloated to climb at a vertical angle as ants usually can; the steepest it can climb is at an angle to the horizontal of (pi/6). The sugar bowl is shaped like a paraboloid z = x^2 + y^2 , 0 <= z <= 4 where the coordinates are in centimeters.
1. Find the path the ant takes to get to the top of the sugar bowl, assuming it climbs as steeply as possible at each point. Hint: You might use cylindrical coordinates (r,theta,z) or polar coordinates in the xy-plane and think of the (last segment of the) ant's path as parameterized by theta.
2. What is the length of the ant's path from the bottom to the rim?
3. Draw a graph of the sugar bowl and the path the ant takes to get out. Hint: you may want to start with the projection of the path r(theta) in the (r,theta) plane.
1. Find the path the ant takes to get to the top of the sugar bowl, assuming it climbs as steeply as possible at each point. Hint: You might use cylindrical coordinates (r,theta,z) or polar coordinates in the xy-plane and think of the (last segment of the) ant's path as parameterized by theta.
2. What is the length of the ant's path from the bottom to the rim?
3. Draw a graph of the sugar bowl and the path the ant takes to get out. Hint: you may want to start with the projection of the path r(theta) in the (r,theta) plane.