SOLUTION: You have a piece of paper that measures 24 cm by 36 cm. You want to cut out a net that can be folded into a box without a lid. Using whole numbers, find the dimensions of the box w

Algebra ->  Finance -> SOLUTION: You have a piece of paper that measures 24 cm by 36 cm. You want to cut out a net that can be folded into a box without a lid. Using whole numbers, find the dimensions of the box w      Log On


   

Question 1055236: You have a piece of paper that measures 24 cm by 36 cm. You want to cut out a net that can be folded into a box without a lid. Using whole numbers, find the dimensions of the box which gives the maximum volume.
Answer by ikleyn(52781) About Me  (Show Source):
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You have a piece of paper that measures 24 cm by 36 cm. You want to cut out a net that can be folded into a box without a lid.
Using whole numbers, find the dimensions of the box which gives the maximum volume.
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The volume is V = (24-2x)*(36-2x)*x = (24*36 - 60x + 4x^2)*x = 4x^3 - 60x^2 + 864x

Take the derivative %28dV%29%2F%28dx%29:

%28dV%29%2F%28dx%29 = 12x^2 - 120x + 864 = 12*(x^2 - 10x + 72) = (complete the square) = 12*((x-5)^2 + 97).

The maximum volume is at x = 5.

Then the dimensions of the box are 14 cm x 26 cm x 5 cm.

On completing the square see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
in this site.


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.


The referred lessons are the part of this online textbook under the topic "Finding minimum/maximum of quadratic functions".