SOLUTION: Canadian Postal Service regulations require that the sum of three dimensions of a rectangular package not exceed 3 m. What are the dimensions of the largest rectangular box with s

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Question 199523: Canadian Postal Service regulations require that the sum of three dimensions of a rectangular package not exceed 3 m. What are the dimensions of the largest rectangular box with square ends that can be mailed?
Answer by solver91311(24713) About Me  (Show Source):
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I have to presume that you are a calculus student and are familiar with the process of finding the first and second derivatives of a polynomial function because that is the only way I know to solve this problem.

Another assumption that I have to make is that by "largest rectangular box" you mean the "rectangular box with the largest volume".

Given all of that:

The restriction is that the sum of the dimensions be not exceed 3 m and that at least one pair of opposite faces of the rectangular box are squares.

Let represent the measure of the side of one of the square faces. Then we can say that the remaining dimension is .

The volume is the product of the three dimensions so the volume function with respect to the dimension of the square end is:



The feasible domain for in this situation is because if then you obviously have a zero volume box and if then the length must be zero because 2 times 1.5 is 3, and again you have a zero volume box.

We are interested in finding a local maximum of the volume function on the interval (0,1.5).

Take the first derivative:



Set the first derivative equal to zero and solve (Fermat's Theorem)





Hence



which must be excluded because it is not in the feasible interval, or



Now that we know that there is a critical point at we need to apply the second derivative test to determine if it is a maximum, minimum, or a possible inflection point.

Take the second derivative:






Therefore V(1) is a local maximum. That means that the measure of each side of the square end of the box must be 1, and since 3 - 2 = 1, the measure of the length must be 1 as well. The maximum volume rectangular box for a given sum of dimensions is a cube.

John