Question

According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 108 in. Find the dimensions of a carton with square base that is not oversized and has maximum volume.

Answer 1

To find the dimensions of a carton with a square base that is not oversized and has maximum volume, we need to understand the given postal regulations. According to the regulations, a carton is considered oversized if the sum of its height and girth (the perimeter of its base) exceeds 108 inches. The maximum volume of the carton can be achieved by choosing the length of the side of the square base as 21.6 inches.

Step-by-step explanation:

To find the dimensions of a carton with a square base that is not oversized and has maximum volume, we need to understand the given postal regulations. According to the regulations, a carton is considered oversized if the sum of its height and girth (the perimeter of its base) exceeds 108 inches.

Let's assume the length of one side of the square base is x. Since it is a square base, all sides will have the same length.

The girth of the base will be 4 times the length of a side, which is 4x. The height of the carton will also be x.

According to the regulations, the sum of the height and girth should not exceed 108 inches. So we can write the equation as:

x + 4x ≤ 108

Combining like terms, we get:

5x ≤ 108

Dividing both sides by 5, we get:

x ≤ 21.6

Since the dimensions of the carton should not exceed 21.6 inches, we can choose any value less than or equal to 21.6 inches for the length of the side of the square base to ensure the carton is not oversized.

The maximum volume of the carton can be found by using the length of the side of the square base (x) as the dimensions of the base and the height (x) as the height of the carton. So the volume V can be calculated as:

V = length × width × height

V = x × x × x

V = x^3

Therefore, to maximize the volume of the carton, we should choose the length of the side of the square base as 21.6 inches.

Answer 2

18" X 18" X 36"

Step-by-step explanation:

Given a square base container of height h, let a side of the base =s

The volume of the container,
`V= s^2h`

If the sum of its height and girth (the perimeter of its base) equals 108 in

`4s+h = 108\\h = 108-4s`

Substituting h=108-4s into V

`V= s^2(108-4s)`

We are required to determine the maximum volume of such container, first we take the derivative:

`V'(s)= s(216-12s)`

Optimizing:

`s(216-12s)=0\\216-12s=0\\12s=216\\s=216 / 12\\s=18`

Recall that: h = 108-4s

`h = 108-4(18)=108-72=36inch`

The dimensions of the carton are 18" X 18" X 36".