1)It has been conjectured that a fish swimming a distance of L ft at a speed of V ft/sec relative to the water and against a current flowing at the rate of U ft/sec (u<v) expends a total energy given by: E(v)= aLv^3/v-u

where E is measured in foot-pounds (ft-lb) and a is a constant. Find the speed V at which the fish must swim in order to minimize the total energy expended.

2) By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 15in long and 8 in wide, find the dimensions of the box that will yield the maximum volume,

3) In an open box has a square base and volume of 108 cubic inches, and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amoun o material is used in construction.

4) A grain silo has the shape of a right circular cylinder surronded by a hemisphere. If the Silo is to have a capacity of 504*PI cubic feet, find the radius and height of the silo that requires the least amount of material to construct.
(The volume of the silo is pi*r^2*h + 2/3*pi*r^3, and the surface area, including the floor, is pi(3*r^2 + 2*r*h)

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