Maximum volume using AM GM inequality

In summary, the maximum volume of a carry-on bag for an airline's requirements is 27000 cubic centimeters. This can be calculated by finding the maximum value for length, width, and height, all being 30 cm, resulting in a volume of 27000 cm^3. Using physical units is important when giving answers in problems involving them.
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Hi everyone,

I'm a bit confused with this question.

An airline demands that all carry-on bags must have length + width + height at most 90cm. What is the maximum volume of a carry-on bag? How do you know this is the maximum?

[Note: You can assume that the airline technically mean "all carry on bags must fit inside some rectangular prism with length + width + height at most 90cm". Remember that the volume of a rectangular prism is given by length x width x height.]

My attempt at the question:

View attachment 2497I thought my answer was to big for a volume. Any help would be greatly appreciated!
 

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  • #2
I have moved this thread since this is a better fit.

Your answer looks correct to me (in $\text{cm}^3$), as I find the same value using cyclic symmetry, which implies the maximum will occur for:

\(\displaystyle \ell=w=h=30\text{ cm}\)
 
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Thanks!
 
  • #4
One way of looking at this is that a cubic centimetre is a very small volume. If you had given the result in cubic metres then it would have been $0.027\,\text{m}^3$, and you might have thought that the answer was too small.

In problems that use physical units, you should always specify the units when giving the answer.
 
  • #5
That is true, I probably would have thought it was too small if the units was in m^3. Thanks!
 

FAQ: Maximum volume using AM GM inequality

What is the AM GM inequality and how is it used to find maximum volume?

The AM GM inequality, also known as the Arithmetic Mean-Geometric Mean inequality, states that for a set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean. This inequality can be used to find the maximum volume of a rectangular prism by determining the side lengths that will result in the largest possible product of the lengths.

What is the formula for finding maximum volume using the AM GM inequality?

The formula for finding maximum volume using the AM GM inequality is V = (a^3)(b^3)(c^3), where a, b, and c are the side lengths of the rectangular prism. This formula can be derived by using the AM GM inequality to find the maximum product of the side lengths, which will result in the maximum volume.

What are the steps for using the AM GM inequality to find maximum volume?

The steps for using the AM GM inequality to find maximum volume are:
1. Identify the variables given in the problem, which will represent the side lengths of the rectangular prism.
2. Use the AM GM inequality to set up an equation that relates the variables and their corresponding arithmetic and geometric means.
3. Solve the equation for the maximum value of the product of the side lengths.
4. Substitute this maximum value into the formula for volume (V = (a^3)(b^3)(c^3)) to find the maximum volume.

Can the AM GM inequality be used for any shape to find maximum volume?

No, the AM GM inequality can only be used for finding maximum volume of rectangular prisms. This is because the inequality only holds true for positive numbers and the formula for volume of a rectangular prism (V = lwh) involves multiplication of three positive numbers.

Are there any limitations or conditions for using the AM GM inequality to find maximum volume?

Yes, there are a few limitations and conditions for using the AM GM inequality to find maximum volume:
- The inequality can only be used for finding maximum volume of rectangular prisms.
- All side lengths must be positive numbers.
- The inequality may not always result in the exact maximum volume, but it will give a close approximation.
- It is important to check the solutions obtained from the inequality to ensure they make sense in the context of the problem.

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