# Maximizing Box Volume with 1200cm^2 of Material

• Apost8
In summary, the conversation discusses a math problem involving finding the maximum volume of a box with a square base and an open top given a limited amount of material. The solution involves writing equations for the area and volume of the box, solving for one variable in terms of the other, and using differentiation to find the maximum volume. The conversation ends with the person finding the solution and thanking the others for their help.
Apost8
OK, I've been killing myself over this one problem and I just cannot seem to get it. I know it's probably a lot easier than I'm making it out to be. If anyone can give me a little help I would really appreciate it. Here's the question:

If 1200cm^2 of material is available to make a box with a square base and an open top, what is the box's larget possible volume?

So far, I'm guessing that since the base of the box is to be square and there's an open top, the area of the box = x^2 + 4xy = 1200 and the volume = (x^2)y.

I know at some point I'll need to differentiate and find the maximum of the function, but, I'm sort of floundering at this point. Thanks in advance.

Okay so you know have two functions, one of which you wish to maximise (i.e. the volume). Using your first equation can you write a function for y in terms of x?

So, if I solve the first equation for y, I get y = (1200-x^2)/4x Is this correct?

Apost8 said:
So, if I solve the first equation for y, I get y = (1200-x^2)/4x Is this correct?
Indeed it is. Can you guess what you need to do with this result?

Plug that into my equation for volume, differentiate, and find the max?

Apost8 said:
Plug that into my equation for volume, differentiate, and find the max?
Sounds good to me.

Got it. That wasn't so hard, jeez. Thanks for the help! :)

Apost8 said:
Got it. That wasn't so hard, jeez. Thanks for the help! :)
No worries . Don't forget to verify that your answer is a maximum and is meaningful.

Last edited:

## 1. How do I calculate the maximum volume of a box with 1200cm^2 of material?

To calculate the maximum volume of a box with 1200cm^2 of material, you will need to use the formula V = lwh, where V is the volume, l is the length, w is the width, and h is the height. You will also need to use the formula SA = 2lw + 2lh + 2wh, where SA is the surface area. Set the SA equal to 1200cm^2 and solve for one of the variables (l, w, or h). Substitute this value into the volume formula and solve for the maximum volume.

## 2. What is the optimal shape for maximizing box volume with 1200cm^2 of material?

The optimal shape for maximizing box volume with 1200cm^2 of material depends on the dimensions of the box. Generally, a cube or a rectangular prism with equal sides will have the maximum volume. However, if you are constrained by certain dimensions, a different shape may be more optimal.

## 3. Can I use 1200cm^2 of material to make multiple boxes with maximum volume?

Yes, you can use 1200cm^2 of material to make multiple boxes with maximum volume. However, the number of boxes and their dimensions will depend on the size and shape of each box. To determine the dimensions, you will need to use the same formulas mentioned in the first question.

## 4. What are some real-world applications of maximizing box volume with limited materials?

Maximizing box volume with limited materials is a common problem in packaging and shipping industries. It can also be used in optimizing storage space in warehouses or designing efficient containers for transporting goods. Additionally, this concept can be applied in architecture and construction to make the most of limited building materials.

## 5. Are there any limitations or constraints when maximizing box volume with 1200cm^2 of material?

There may be some limitations or constraints when maximizing box volume with 1200cm^2 of material, such as the dimensions of the box or the type of material being used. Additionally, practical limitations may also play a role, such as the need for the box to be sturdy or easily stackable. It is important to consider these factors when trying to achieve maximum volume with limited materials.

• Calculus and Beyond Homework Help
Replies
11
Views
4K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
17
Views
4K
• Calculus and Beyond Homework Help
Replies
1
Views
889
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
4K
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
978