# Optimization of box, varied material cost.

• QuarkCharmer
In summary, the conversation discusses finding the cost of materials for the cheapest rectangular storage container with a volume of 10m³. The length of the base is twice the width and the cost of materials for the base and sides are $10 per square meter and$6 per square meter, respectively. The formula for the cost of the box is given as C(W) = (180+20W³)/W and the minimum cost is found when the width is 1.65, resulting in a cost of approximately $163.50. QuarkCharmer ## Homework Statement Stewart Calculus 6E: 4.7 #14 A rectangular storage container with an open top is to have a volume of 10m³. The length of it's base is twice the width. Material for the base costs$10 per square meter. Material for the sides cost $6 per square meter. Find the cost of materials for the cheapest such container. ## Homework Equations ## The Attempt at a Solution I let the width of the box be W, the length be 2W, and the height be H. Since: $$2W^{2}H=10$$ I let $H=\frac{10}{2W^{2}}$ Then I claim that the cost of the base is given by: $$(2W^{2})10 = 20W^{2}$$ The cost of the sides are given by: $$(2WH + 4WH)6$$ So the total cost for the box could be written as: $$20W^{2} + (2WH + 4WH)6$$ Substituting in $H=\frac{10}{2W^{2}}$, I get cost as a function of width? $$C(W) =(2W(\frac{10}{2W^{2}})+4W(\frac{10}{2W^{2}}))6 + 20W^{2}$$ $$C(W) = \frac{180+20W^{3}}{W}$$ So I can minimize that function, and I find that the minimum is when the width is 1.65, and the cost of the box is about$163.50. So \$163.50 is the solution? Does anyone see a problem with this?

Nope, that's right. :)

## 1. What is optimization of box?

Optimization of box refers to the process of finding the most efficient and cost-effective way to design and manufacture a box. This involves considering factors such as the size, shape, material, and cost of the box.

## 2. Why is optimization of box important?

Optimization of box is important because it can significantly impact the overall cost and quality of a product. By finding the most optimal design, companies can save money on materials and production costs, while also ensuring that the box effectively protects and presents the product.

## 3. How is the material cost of a box determined?

The material cost of a box is determined by the type and quantity of materials used. This can include factors such as the type of material (cardboard, plastic, etc.), the thickness or weight of the material, and any additional features such as printing or coating.

## 4. What are some methods for optimizing box material cost?

Some methods for optimizing box material cost include using lightweight materials, minimizing the amount of material waste, and considering alternative materials that may be more cost-effective. Additionally, using software or simulations to test different designs and material combinations can also help in finding the most cost-efficient option.

## 5. What are the benefits of optimizing box material cost?

The benefits of optimizing box material cost include reducing production costs, improving profit margins, and potentially lowering the final cost for consumers. It can also lead to more sustainable packaging solutions by minimizing material waste and using eco-friendly materials.

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