Optimization Suppose postal requirements are that the maximum of the length plus the girth (cross sectional perimeter) of a rectangular package that may be sent is 275 inches. Find the dimensions of the package with square ends whose volume is to be maximum. (Leave your answers to 3 decimal places.) Square side :Length 

Expert Answers

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You should come up with the following substitution for length of side of square such that:

l=x

h represents the height of the package

The problem provides the following information about postal requirements such that:

275 = h + 2(h + xsqrt2)

Notice that 2(h + xsqrt2) represents the cross sectional perimeter of the package.

Opening the brackets yields:

275 = 3h + 2sqrt2*x

You need to evaluate the volume of rectangular package such that:

V = x^2*h

You should use the equation that expresses the postal requirement to write h in terms of x such that:

`3h = 275 - 2sqrt2*x => h = (275 - 2sqrt2*x)/3`

You need to substitute `(275 - 2sqrt2*x)/3 ` for h in equation of volume such that:

`V = x^2*(275 - 2sqrt2*x)/3`

You need to optimize the volume, hence, you need to differentiate the function of volume with respect to x such that:

`V'(x) = 2x*(275 - 2sqrt2*x)/3- 2sqrt2*x^2/3`

`V'(x) = 550x/3 - 4sqrt2*x^2/3 - 2sqrt2*x^2/3`

`V'(x) = 550x/3 - 6sqrt2*x^2/3`

You need to solve for x the equation `V'(x) = 0`  such that:

`V'(x) = 0 => 550x/3 - 6sqrt2*x^2/3 = 0`

`550x - 6sqrt2*x^2 = 0`

You need to factor out `2x`  such that:

`2x(275 - 3sqrt2*x) = 0 => x = 0 ` invalid

`275 - 3sqrt2*x = 0 => x = 275/(3sqrt2) => x = 275sqrt2/6`  inches

Hence, evaluating the dimension that meets the postal requirements yields `x = 275sqrt2/6`  inches.

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