A rectangular storage container with an open top is to have a volume of $14$ cubic meters. The length of its base is twice the width. Material for the base costs $10$ dollars per square meter. Material for the sides costs $5$ dollars per square meter. Find the cost of materials for the cheapest such container.

Total cost $=($Round to the nearest penny and include monetary units. For example, if your answer is $1\mathrm{.}095,$ enter $$1\mathrm{.}10$ including the dollar sign and second decimal place.$)$