Question

You need to construct an open-top rectangular box with a base that has a length that is 3.5 times the width and that must hold a volume of exactly 825 cm3. The material for the base of the box costs 8 cents/cm2 and the material for the sides of the box costs 7 cents/cm2.
Find the dimensions for the box that will minimize the cost of the materials used to construct box. (Remember, you are not supposed to use calculus to do this.)

Be sure you define all variables you use and then, graph using your calculator to answer the question.

Answer 1

The dimensions for this rectangular box which would minimize the cost of the materials used to construct box are:

1. Length = 22.488 cm.
2. Width = 6.425 cm.
3. Height = 5.710 cm.

How to determine the dimensions of this box?

First of all, we would assign variables to the dimensions of this rectangular box and then calculate its height as follows:

• Let x represent the width of the base of this rectangular box.

• Let L represent the length of this rectangular box.

• Let H represent the height of of this rectangular box.

• Let the total cost of this rectangular box be y.

Therefore, its length would be given by 3.5x.

How to calculate the volume of a rectangular box?

Mathematically, the volume of a rectangular box should be calculated by using this formula:

V = L × x × H

H = V/(Lx)

H = 825/(3.5x × x)

H = 1650/7x² cm.

Next, we would determine the cost of of the base and side of this rectangular box as follows:

Base cost = 0.08 × Lx

Base cost = 0.08 × 3.5x²

Base cost = $0.28x²

For the side, we have:

Cost of side = 0.07 × H × (2 × (L + x))

Cost of side = 0.14(1650/7x² × (x + 3.5x)

Cost of side = $148.5/x

Therefore, the total cost of this rectangular box is given by:

Total cost = Base cost + Cost of side

Total cost = 0.28x² + 148.5/x

By using Geogebra online graphing calculator, we would plot a graph of the width of this rectangular box against the total cost as shown in the image attached below.

From the graph, we can logically deduce the following information:

• Width = 6.425 cm.

• Minimum cost = $34.67

For the other dimensions (length and height), we have:

Length = 3.5x = 3.5 × 6.425 = 22.488 cm.

Height = 1650/7x² = 1650/(7 × 6.425²) = 5.710 cm.

Read more on rectangular box here: brainly.com/question/3867601

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Answer 2

Answer:

• width: 6.425 cm
• length: 22.488 cm
• height: 5.710 cm

Step-by-step explanation:

We want to find dimensions of an open-top box of volume 825 cm³ such that cost of materials is minimized. The length relative to the width is specified, as are the costs of side and bottom materials.

Setup

Variables and dimensions

Let x represent the width of the base of the box. Then the length is 3.5x. The height is found from ...

  V = LWH

  H = V/(LW) = 825 cm³/(x(3.5x)) = 1650/(7x²)

So, the dimensions of the box are ...

• width: x
• length: 3.5x
• height: 1650/(7x²)

Cost of materials

The cost of materials for the base is $0.08/cm². The area of the base is the product of length and width.

  base cost = $0.08 × LW = $0.08 × (3.5x³) = $0.28x²

The cost of materials for the sides is $0.07/cm². The total area of the sides of the box is the product of the height and the perimeter of the base.

  side cost = $0.07 × (H)(2(L+W)) = $0.14(1650/(7x²)(x +3.5x) = 148.5/x

The total cost (y) is the sum of the base cost and the side cost. This is the value we want to minimize.

  total cost = 0.28x² +148.5/x

Solution

Graph

A graph of the cost versus the width of the box is attached. It shows the minimum cost to be about $34.67 when the width of the box is 6.425 cm.

Dimensions

The remaining dimensions of the box are ...

  length = 3.5(6.425 cm) = 22.488 cm

  height = 1650/(7·6.425²) cm = 5.710 cm

  width = 6.425 cm

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Additional comment

You will find that the cost of the base is half the cost of the sides. This is the general solution to the problem, and can be used to find the exact dimensions without graphing or calculus.

Unfortunately, that fact cannot be shown definitively without using calculus.