A postal service will accept a package if its length plus its girth is not more than 96 inches. Find the dimensions and volume of the largest package with a square end that can be mailed.

Step-by-step explanation:

Let the length and width of the rectangular package be x and y respectively. Since end of the package is a square, the perimeter of the package will be expressed as P = 4x+y and the volume will be expressed as V = x²y

If a postal service will accept a package if its length plus its girth is not more than 96 inches, then the perimeter is equivalent to 96 inches.

96 = 4x+y

y = 96-4x

Substituting the value of x into the formula for calculating the volume, we will have;

V(x) = x²(96-4x)

V(x) = 96x²-4x³

To get the dimensions and volume of the largest package, we will find V'(x) and equate it to zero.

V'(x) = 192x-12x²

192x-12x² = 0

Factoring out x;

x(192-12x) = 0

x = 0 and  192-12x = 0

12x = 192

x = 192/12

x = 16

This shows that we have a maximum value at x = 16 and minimum at x = 0

To get y, we will substitute x = 16 into the expression y = 96-4x

y = 96-4(16)

y = 96-64

y = 32

- The dimensions of the largest package is therefore 16in by 16in by 32in

- Volume of largest package = x²y = 16²*18 = 8,192in³

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