Write the following power series in summation (sigma) notation. x cubedminusstartfraction x superscript 5 over 4 endfraction plusstartfraction x superscript 7 over 9 endfraction minusstartfraction x superscript 9 over 16 endfraction plusmidline ellipsis

A1kg cart slams into a stationary 1kg cart at 2 m/s. the carts stick together and move forward at a speed of 1 m/sl. determine whether kinetic energy was conserved in the collision. use the law of conservation of energy to explain the collision

When you drop a pebble from height a it reaches the ground with speed v if there is no air resistance. from what height should you drop it so it will reach the ground with three times the speed? (a) 9m (b) 3m (c) 6m (d) 18m

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.)