The volume of a square based rectangular box is $252$ dm$\text{'}.$ The construction cost of the bottom is$$5\mathrm{.}00$ per dm$?,$ of the top is $$2\mathrm{.}00$ per dm$?$ and of the sides is $$3\mathrm{.}00$ per dm$?.$ Find the dimensionsthat will minimize the cost if the side of the base must fall between $4$ dm and $8$ am$.$