A rectangular storage container with a lid is to have a volume of 8 m. The length of its base is twice the width. Material for the base costs $4 per m. Material for the sides and lid costs $8 per m. Find the dimensions of the container which will minimize cost and the minimum cost.


base width = m

base length = m

height = m

To find the dimensions of the container that will minimize the cost, we need to find the values for the base width, base length, and height that will result in the minimum cost.

Let's start by defining the variables:
- Let x be the base width of the container (in meters)
- Since the length of the base is twice the width, the base length will be 2x (in meters)
- Let h be the height of the container (in meters)

The formula for the volume of a rectangular container is:
Volume = length * width * height

From the given information, we know that the volume is 8 m^3:
8 = 2x * x * h
8 = 2x^2h

Now, let's write the formula for the cost:
Cost = (2 * base area * cost per m) + (3 * side area * cost per m)

The base area is given by:
Base Area = length * width = 2x * x = 2x^2

The side area is given by:
Side Area = 2 * (length * height + width * height) = 2 * (2x * h + x * h) = 2 * (3xh) = 6xh

The cost formula becomes:
Cost = (2 * 2x^2 * 4) + (3 * 6xh * 8)
Cost = 16x^2 + 144xh

We need to express the cost formula in terms of one variable, so we'll substitute the value of volume we found earlier:
h = 8 / (2x^2)
h = 4 / x^2

Now, substitute h into the cost formula:
Cost = 16x^2 + 144x(4 / x^2)
Cost = 16x^2 + 576 / x

To find the minimum cost, we need to take the derivative of the cost function with respect to x and set it equal to zero. Let's do that:

dCost/dx = 32x - 576 / x^2

Setting it equal to zero:
32x - 576 / x^2 = 0

Multiply both sides by x^2 to get rid of the denominator:
32x^3 - 576 = 0

Solving for x^3:
x^3 = 576 / 32
x^3 = 18
x = ∛18
x ≈ 2.6207

Now that we have the value of x, we can substitute it back into the equation for h:
h = 4 / x^2
h = 4 / (2.6207^2)
h ≈ 0.5843

Therefore, the dimensions of the container that will minimize the cost are approximately:
Base width ≈ 2.6207 meters
Base length ≈ 5.2414 meters
Height ≈ 0.5843 meters

To find the minimum cost, substitute these values back into the cost formula:
Cost = 16x^2 + 144xh
Cost ≈ 16(2.6207)^2 + 144(2.6207)(0.5843)
Cost ≈ 141.6008

Therefore, the minimum cost for the container is approximately $141.60.