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A rectangular box with a square base and no top is to be made of a total of 120 cm2 of cardboard. Find the dimensions of the box of maximum volume.

User Roudy
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Answer:

The dimensions of the box of maximum volume are: Length: 6.325 centimeters, height: 3.162 centimeters.

Explanation:

The volume (
V), measured in cubic centimeters, and the surface area of the rectangular box (
A_(s)), measured in square centimeters, with a square base are represented by the following formulas:


V = l^(2)\cdot h (1)


A_(s) = 4\cdot l\cdot h +l^(2) (2)

Where:


l - Length of the side of the square base, measured in centimeters.


h - Height of the box, measured in centimeters.

By (2) we clear the height:


4\cdot l\cdot h = A_(s) -l^(2)


h = (A_(s)-l^(2))/(4\cdot l)

And expand (1) by this result:


V = l^(2)\cdot \left((A_(s)-l^(2))/(4\cdot l ) \right)


V = (A_(s)\cdot l -l^(3))/(4)

Lastly, we proceed to perform First and Second Derivative Test to find critical values associated with maximum volume:

First Derivative Test


(A_(s)-3\cdot l^(2))/(4) = 0


A_(s) = 3\cdot l^(2)


l =\sqrt{(A_(s))/(3) } (1)

Second Derivative Test


V''=-(6\cdot l)/(4)


V'' = -(3\cdot l)/(2)


V'' = -(3)/(2)\cdot \sqrt{(A_(s))/(3) } (2)

We conclude that critical value leads to a maximum volume.

If we know that
A_(s) = 120\,cm^(2), then dimensions of the box are, respectively:


l = \sqrt{(120\,cm^(2))/(3) }


l \approx 6.325\,cm


h = (120\,cm^(2)-(6.325\,cm)^(2))/(4\cdot (6.325\,cm))


h \approx 3.162\,cm

The dimensions of the box of maximum volume are: Length: 6.325 centimeters, height: 3.162 centimeters.

User Alvin SIU
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