What dimensions should a rectangular open-top storage container have to achieve a volume of 10 m³ if the length of the base is twice the width, and how much will the material for the base cost?

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Calculating the Dimensions and Cost of the Storage Container

To determine the dimensions of the rectangular storage container with an open top, we need to use the given conditions:

  • The volume of the container is 10 m³.
  • The length of the base is twice the width.
  • The material cost for the base is $15 per square meter.

Let’s denote:

  • w = width of the base (in meters)
  • l = length of the base (in meters) = 2w
  • h = height of the container (in meters)

Using the volume formula for a rectangular prism:

Volume = length × width × height
10 = l × w × h

Since we know that l = 2w, we can substitute this into the volume equation:

10 = (2w) × w × h
10 = 2w²h

We can rearrange this to find the height:

h = 10 / (2w²) = 5 / w²

Next, we need to find the area of the base to calculate the cost of the material:

Area of base = length × width = l × w = (2w) × w = 2w²

Now, the cost of the material for the base can be calculated as:

Cost = Area × Cost per square meter = 2w² × 15
Cost = 30w²

To find the optimal dimensions, we can express the volume condition in terms of w. We have two equations now:

  • Height: h = 5 / w²
  • Cost: Cost = 30w²

Next, let’s calculate the total cost for different values of width:

Assuming you want a minimum cost, we can use Calculus to find the critical points. However, for simplicity, let’s assume reasonable dimensions:

Suppose w = 1 meter:

  • Length = 2w = 2 × 1 = 2 meters
  • Height = h = 5 / (1)² = 5 meters
  • Area of base = 2 × (1)² = 2 m²
  • Cost of the base = 2 × 15 = $30

Now, let’s try w = 2 meters:

  • Length = 2w = 2 × 2 = 4 meters
  • Height = h = 5 / (2)² = 5 / 4 = 1.25 meters
  • Area of base = 2 × (2)² = 8 m²
  • Cost of the base = 8 × 15 = $120

Finding a suitable width to minimize cost while maintaining the volume constraint can be achieved by testing values. In practice, you may need to evaluate several widths until you find the optimal size that balances material cost and volume.

Conclusion: From our calculations, if you choose a width of 1 meter, the dimensions of the container would be 2 meters in length, 1 meter in width, and 5 meters in height, with a base material cost of $30.

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