A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $20 per square meter. Material for the sides costs $12 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)$

Sure, let's solve this step by step.

Step 1: Define the variables Let's denote the width of the base as x (in meters), and the length as 2x (since it's twice the width). The height of the container will be h (in meters).

Step 2: Express the volume in terms of x and h The volume V of a rectangular prism is given by the formula V = length * width * height. In this case, V = 2x * x * h = 10 m^3. From this, we can express h in terms of x: h = 10 / (2x^2) = 5 / x^2.

Step 3: Express the cost in terms of x The cost C of the materials is the sum of the cost of the base and the cost of the sides. The base has an area of 2x * x = 2x^2, so its cost is 20 * 2x^2 = 40x^2. Each side has an area of x * h or 2x * h, so the total cost of the sides is 2 * 12 * (x * h + 2x * h) = 24xh + 48xh = 72xh. Substituting h = 5 / x^2, we get C = 40x^2 + 72x * (5 / x^2) = 40x^2 + 360 / x.

Step 4: Minimize the cost To find the minimum cost, we need to find the minimum of the function C(x) = 40x^2 + 360 / x. This can be done by taking the derivative of C with respect to x, setting it equal to zero, and solving for x. The derivative is C'(x) = 80x - 360 / x^2. Setting this equal to zero gives 80x = 360 / x^2, or x^3 = 360 / 80 = 4.5. Taking the cube root of both sides gives x = 1.65 m (rounded to two decimal places).

Step 5: Calculate the minimum cost Substituting x = 1.65 into the cost function gives C = 40 * (1.65)^2 + 360 / 1.65 = $165.45 (rounded to the nearest cent).

So, the cost of materials for the cheapest such container is $165.45.