There is a traditional problem (your text has an example in Exercise 10) that go

Question

There is a traditional problem (your text has an example in Exercise 10) that goes like this: "We want to make an open-topped box from an 8.5 times 11 inch sheet of paper by cutting congruent squares from the comers and folding up the sides. What is the maximum possible volume of such a box?" What most people never think about is the fate of those four squares of paper. They don't have to be wasted By taping them together, and putting the resultant structure on a desk, one can make a handsome pen-and-pencil holder, which will be a box with neither top nor bottom. (It will still hold pencils as long as it rests on the desk.) What is the maximum possible combined volume of an open-topped box plus a handsome pen-and-pencil holder that can Sc made by cutting four squares from an 8.5 times 11 inch sheet of paper? Describe the open-topped box that results from this maximal case. Intuitively, why do we get the result we do? Repeat this problem for a 6 times 10 inch piece of paper.

Explanation / Answer

1.

Volume of box with bottom = (8.5-2x)*(11-2x)*x
                                                            = (93.5 - 17x -22x + 4x^2)*x
                                                            = (93.5*x - 17*x^2 -22*x^2 + 4x^3)
                                                            = (93.5*x - 39*x^2 + 4x^3)
volume of both with no bottom = x*x*x = x^3
Total volume,
V = (93.5*x - 39*x^2 + 4x^3) + x^3
    = (93.5*x - 39*x^2 + 5x^3)

dV/dx = 93.5 - 78*x + 15*x^2
put dV/dx = 0
93.5 - 78*x + 15*x^2 = 0
solving above quadratic equation we get,
x = 3.3 and x = 1.9

d^2V/dx^2 = -78 + 30*x
at x = 3.3,
d^2V/dx^2 = -78 + 30*3.3 = 21
since d^2V/dx^2 is positive, at x = 3.3 we get minimum volume

at x = 1.9,
d^2V/dx^2 = -78 + 30*1.9 = -21
since d^2V/dx^2 is negative, at x = 1.9 we get maximum volume

so,
maximum volume = (93.5*x - 39*x^2 + 5x^3)
                                      = (93.5*(1.9) - 39*(1.9)^2 + 5*(1.9)^3)
                                      = 71.2 in^3

2)
I dont know what description it is asking about but I am sure you cana nswer it

3)
please solve the 1 again .
Its simple
Let me know if I need to do the repetive work solving it again

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