the middle of the tank is a cylinder whose length is 10 feet and there is a hemisphere attached to each end. Determine the radius x so that the volume of the tank is 27 pi ft^3
how is this solved??

The volume of the tank is equal to the sum of the volume of the central cylinder plus the two hemispherical ends. Express these volumes in terms of the unknown radius R, then set the total volume to 27 pi ft^3, and solve for R. You'll find that you need to solve a cubic - a little trial an error will let you find a solution to that pretty easily.