## rebootkz 3 years ago Optimization Problems....! =='' Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.

1. rebootkz

Write an expression for the volume V in terms of x and y.

2. rebootkz

for me the problem is can't understand the situation..

3. rebootkz

|dw:1340359906805:dw|

4. rebootkz

am I right or not? and then?

5. dg123

u r wrong

6. dg123

|dw:1340360018176:dw|

7. dg123

the shaded region will be folded to form the cuboid

8. dg123

is the answer 1 cubic feet?

9. rebootkz

where is y and x? don't know the answer yet.. you mean the largest volume of such situation should be 1 cubic ft ?

10. dg123

yes, it is to be solved by AM-GM inequality

11. rebootkz

2 ft ^3

12. dg123

hw cum?

13. vood

I think $V=y \times y \times x =y ^{2}x$

14. rebootkz

|dw:1340360932310:dw|

15. rebootkz

yep V=xy^2 2x+y=3

16. ganeshie8

@rebootkz is this in calculus chapter ?

17. rebootkz

yep ...==

18. rebootkz

math 1 A

19. ganeshie8

then it is easy

20. dg123

sir i still think the answer is 1 cubic feet

21. ganeshie8

write out an equation for V :

22. rebootkz

V = (3 - 2x)^2(x) = (9 - 12x + 4x^2)(x) = 9x - 12x^2 + 4x^3 V'=3(2X-3)(2X-1)

23. ganeshie8

V' = 0, and find out X values

24. rebootkz

X=1/2 OR 3/2 V(1/2)=2 V(3/2)=0 SO IT'S 2

25. rebootkz

@dg123 got it ??

26. ganeshie8

yep you have it :)

27. dg123

oh the balls on me,,,, i found the minimum volume :(

28. rebootkz

yeppp ! thanks :DDD @dg123 @ganeshie8 @vood

29. rebootkz

okkk !! hahaa never mind ! best answer ! hahahaha thx "D

30. dg123

hmm ha ha :)

31. ganeshie8

lol.. why you left X = 3/2 ?

32. ganeshie8

oh ok.. you have shown it in the above equation... im seeing now only lol

33. rebootkz

V = (3 - 2x)^2(x) = (9 - 12x + 4x^2)(x) = 9x - 12x^2 + 4x^3 V' = 9-24x+12x^2=3(4x^2-8x+3)=3(2X-3)(2X-1) LET V'(X)=0 THEN ...

34. ganeshie8

its not : let V'(X) = 0 to find where the Volume has extreme values, we are equating V' (slope of Volume function) = 0

35. rebootkz

yep , exactly ! anyways, i got the answer ! haha

36. ganeshie8

|dw:1340361872423:dw|

37. ganeshie8

ok cool..

38. rebootkz

my hw is due 7:30am ...very soon.. goshhh! I still have TWo chapters to do !!!! so ... just do simple way...!!!! thx :DDDD

39. ganeshie8

okk run... :)

40. dg123

sprnit fast :)

41. rebootkz

lol hopefully...!

42. vood

I am not sure but i think that the maximum volume is 27 cubic feet

43. ganeshie8

lol..

44. ganeshie8

@vood must be jk.. we need four times the material to have that volume..

45. holls622

(e) Use part (d) to write the volume as a function of x.