Final answer:
The cost of materials for the storage container is a function of the width of the base. Using the volume and the cost of materials for the base and sides, and simplifying the equations, the cost is found to be (40w + 80) dollars, where w is the width.
Step-by-step explanation:
The cost of materials for the rectangular storage container with an open top can be expressed as a function of the width of the base. To find this function, let's denote the width of the base as w, the length as 2w (since it is twice the width), and the height as h. Given the volume of 10^3 cubic units, we have w * 2w * h = 10^3. From this, we can solve for h as h = 10^3 / (2w^2).
Now, we calculate the cost of materials:
The base cost is 10 dollars per square meter, so the cost is 10 * w * 2w.
The sides cost is 4 dollars per square meter. There are four sides, their areas being 2wh, wh, 2wh, and wh respectively. The total area of the sides is 2(2wh + wh).
Substituting the value of h, the side cost becomes 4 * 2(2w * (10^3 / (2w^2)) + w * (10^3 / (2w^2))) = 40w + 80. Therefore, the cost of materials as a function of the width w is 20w^2 + 40w + 80, which simplifies to 40w + 80 dollars. The correct answer is (a) (40w + 80).