Final answer:
The cost of materials for the cheapest container is approximately 51.1 dollars. To find the cost of materials for the cheapest rectangular storage container, we need to determine the dimensions that minimize the cost. We can solve this problem by setting up equations for the volume and cost of materials, and then differentiate the cost function with respect to one of the variables to find the critical point. By substituting the values back into the cost function, we can find the cost of materials for the cheapest container.
Step-by-step explanation:
To find the cost of materials for the cheapest rectangular storage container with a volume of 12 cubic meters, we need to determine the dimensions of the container that minimize the cost. Let's assume the width of the base is x meters. Since the length of the base is twice the width, the length would be 2x meters. The height of the container doesn't affect the cost of materials.
The volume of the container can be calculated as V = length * width * height. In this case, V = 2x * x * h = 12, where h is the height of the container.
Since the height doesn't affect the cost, we can solve for x in terms of h: 2x * x * h = 12, which simplifies to x²h = 6.
Now, let's express the cost of materials in terms of x and h. The cost of the base would be 13 * (2x * x) = 26x² dollars, and the cost of the sides would be 4 * (2x * h) = 8xh dollars.
Substituting x²h = 6 into the cost equations, we get a cost function: C = 26x² + 8xh. To minimize the cost, we can use calculus or trial and error. Solving for x in terms of h using the volume equation, we get x = sqrt(6/h). Substituting this into the cost function, we get C = 26(sqrt(6/h))^2 + 8(sqrt(6/h))h.
Now, we can differentiate the cost function with respect to h and set it equal to zero to find the critical point: dC/dh = -26(sqrt(6/h))^2/h + 8(sqrt(6/h)) = 0. Solving this equation, we find h ≈ 1.79.
Substituting the value of h back into the volume equation, we find x ≈ 1.66. Finally, substituting the values of x and h into the cost function, we get C ≈ 51.1 dollars. Therefore, the cost of materials for the cheapest container is approximately 51.1 dollars.