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...
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
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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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