According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 192 in. Find the dimensions of a carton (in inches) with square base that is not oversized and has maximum volume. (Enter the three dimensions as a comma-separated list.)

Question

Mathematics

Kaylee Green

3 September, 02:53

According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 192 in. Find the dimensions of a carton (in inches) with square base that is not oversized and has maximum volume. (Enter the three dimensions as a comma-separated list.)

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Answers (1)

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Andre Crawford · Answer 1

Square base dimension = 15 inches

Maximum volume = 7200 inches^3

Step-by-step explanation:

V = x^2y ... eq 1

Let the square base be x and the height y

Oversize formular is given by

Y + 4x = 92

Y = 92 - 4x ... eq 2

Put eq 2 into eq 1

V = x^2 (92 - 4x^3)

V = 92x^2 - 4x^3

Using derivatives

V = 184x - 12x^2

V' = 0 = 184x - 12x^2

X (184 - 12x)

X=0

X = 184/12 = 15.33 approximately 15 inches

Maximum Volume = V = 92 (15) ^2 - 5 (15) ^3

V = 92 (225) - 4 (3475)

V = 20700 - 13500

V = 7200 inches^3