Surface Area of Shoe Box Shape to Maximize Volume

In summary, the problem presents a rectangular piece of cardboard measuring 3 feet by 4 feet and asks for the outside surface area of a shoe box shape that maximizes the volume when four squares are cut out from the corners. The equations used are V = L*W*H and 2ab + 2bc + 2ac. The approach is to express the length, width, and height in terms of x after removing the squares, and then determining the height of the resulting flaps.
  • #1
JuliusDarius
25
0

Homework Statement


Imagine you have a rectangular piece of cardboard measuring 3 feet by 4 feet. You know that if you cut a square out of each corner, you can fold the pieces together and tape them together to make an object that looks like a shoe box:http://www.omahamathtutor.com/wp-content/uploads/2012/03/shoebox.png
What is the outside surface area of this shoe box shape that maximize the volume?



Homework Equations


2ab + 2bc + 2ac


The Attempt at a Solution


Not sure where to start
 
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  • #2
V = L*W*H

Then express the length, width, and height in terms of x after you remove those four squares from the 4 x [STRIKE]12[/STRIKE] 3 rectangle.
 
Last edited:
  • #3
Bohrok said:
V = L*W*H

Then express the length, width, and height in terms of x after you remove those four squares from the 4x12 rectangle.

Could you show me how to do that?
 
  • #4
The length is originally 4, then you cut off two segments of length x from both ends of that side, so L = 4 - 2x. Same thing for the width.

After cutting out the four squares, you have four flaps that fold up; what would be the height of these flaps?
 

What is the surface area of a shoe box shape?

The surface area of a shoe box shape is the total area of all the exposed sides of the box. It can be calculated by finding the area of each side and adding them together.

How can the surface area of a shoe box shape be measured?

The surface area of a shoe box shape can be measured using a ruler or measuring tape. The length, width, and height of the box should be measured and then multiplied together to find the surface area.

Why is maximizing the surface area of a shoe box shape important?

Maximizing the surface area of a shoe box shape is important because it allows for more space inside the box. This is especially useful when packing items into the box, as it allows for more items to fit inside.

How can the surface area of a shoe box shape be increased?

The surface area of a shoe box shape can be increased by increasing the dimensions of the box, such as making it longer or wider. Adding additional flaps or folds to the box's design can also increase the surface area.

What is the relationship between surface area and volume in a shoe box shape?

The surface area and volume of a shoe box shape have an inverse relationship. This means that if the surface area is increased by making the box larger, the volume will also increase. Similarly, if the surface area is decreased by making the box smaller, the volume will decrease.

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