A rectangular storage container with an open top is to have a volume of 10 m3. The length of the base is twice the width. Material for the base is thicker and costs $13 per square meter and the material for the sides costs $10 per square meter. Find the cost of the materials for the cheapest such container. Round to the nearest cent.

the base has area 2w^2

The volume is 10, so the height h=5/w^2

So, the cost is
13(2w^2) + 10(wh + 2wh)
= 26w^2 + 10w(3h)
= 26w^2 + 10w(5/w^2)
= 26w^2 + 50/w

To find the cost of materials for the cheapest container, we need to minimize the cost of the material.

Let's start by determining the dimensions of the container.

Given that the volume of the container is 10 m³, and the length of the base is twice the width, we can set up an equation to represent the volume:

Length × Width × Height = 10

Since the container has an open top, we can assume that the height is equal to the width. Therefore, we have:

Length × Width × Width = 10

Now let's express the length in terms of the width:

Length = 2 × Width

Substituting this into the equation:

2 × Width × Width × Width = 10

Simplifying:

2 × Width³ = 10

Dividing both sides by 2:

Width³ = 5

Taking the cube root of both sides:

Width = ∛5 ≈ 1.71 meters

Since the length is twice the width, we have:

Length = 2 × 1.71 ≈ 3.41 meters

Now that we know the dimensions of the container, we can calculate the cost of the materials.

The base area of the container is Length × Width, and the side area is (Length × Height) + (Width × Height). The height is equal to the width.

Base Area = 3.41 × 1.71 ≈ 5.83 m²
Side Area = (3.41 × 1.71) + (1.71 × 1.71) ≈ 9.10 m²

The cost of the material for the base is $13 per square meter, and the cost for the sides is $10 per square meter.

Cost of Base = 5.83 × 13 = $75.79
Cost of Sides = 9.10 × 10 = $91

Now, we can find the total cost by summing the cost of the base and the sides:

Total Cost = Cost of Base + Cost of Sides
Total Cost = $75.79 + $91 = $166.79

Therefore, the cost of materials for the cheapest container would be approximately $166.79.