Suppose that in a 2-dimensional crystallization process with only one substance, the nuclei do not all appear simultaneously, but rather appear uniformly over space and time at rate λ (nuclei per unit area per unit time). a crystal only starts growing when the nucleus appears. adapt the k-a model to this situation: i. e., find φ(t). hints: work in a 3-dimensional space (2 dimensions of space and one of time. consider an arbitrary point p in this space (arbitrary location, arbitrary time) determine the region, r, in this 3-dimensional space in which a nucleus must appear in order for p to lie inside the circle growing from the nucleus. then follow question 3 above.

Suppose water is leaking from a tank through a circular hole of area ah at its bottom. when water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cah 2gh , where c (0 < c < 1) is an empirical constant. a tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (assume the removed apex of the cone is of negligible height and volume.) (a) suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. the differential equation governing the height h in feet of water leaking from a tank after t seconds is dh dt = − 5 6h3/2 . if the height of the water is initially 8 feet, how long will it take the tank to empty? (round your answer to two decimal places.)