We are looking to build an open rectangular container, without a cover, having capacity $14m^3,$ with the length of its base equal to twice that of its width, as in the diagram above. The material used for the base costs $40$ dollars per square meter, and that used for the walls costs $20$ dollars per square meter. We want to minimize cost.

a$)$ Give a formula for the volume of such a container, as an expression in the variables $x$ and $y$ in the diagram:

Volume $=14m^3=m^3$

b$)$ Give a formula for the cost of building such a container, as an expression in the variables $x$ and $y$ :

Cost $=$ d ${?}_2$ dollars.

Now work out how to minimize the cost of the materials.

c$)$ What are the dimensions of such a container that minimize the cost of materials?

Height of the container $=$

Width of the base $=$ m

FORMATTING: Give the values with an accuracy of at least two decimal places.