algebra

posted by .

open box made from 33 x 33 inch sheet of metal by cutting out x-inch square from each corner and folding up the sides.
What size square should be cut out to produce a box of maximum volume? (This is algebra class)

  • algebra -

    The height of the box becomes x, and each side is reduced by 2x.

    Therefore the volume is given by:

    V = x ( 33-2x )( 33 - 2x )

    because the original length and width are both 33.

    How to find maximum? I don't know .

  • algebra -

    so how do i find the max volume?

  • algebra -

    Not much help from just algebra. If you can't use calculus, your best bet is to graph the function and approximate the value of y at the top of the hump.

  • Algebra -

    1st of all I learned this in my calculus class. Now to the problem
    now since you are going to cut squares (X)from a sheet of 33*33 inches. I am using capital 'X' instead of small
    the volume would be
    V = (33 - 2X)*(33 - 2X)* X
    where one (33 - 2X) is length the other is width and 'X' is height
    then use FOIL or in other words multiply it out
    (4X^2 - 66X - 66X + 1089)* X
    (4X^2 - 132X^2 + 1089)* X
    therefore the final volume equation
    V = (4X^3 - 132X^2 + 1089X)
    now take the derivative of the volume equation
    V' = (12X^2 - 264X + 1089)
    now set the equation equal to zero
    12X^2 - 264X + 1089 = 0
    now you could have factored out 12 from the equation to make it easier but 1089 divided by 12 comes out to a decimal.
    So I used the quadratic formula, so when you plug in the values, I couldn't put in the symbol for square root so just pretend / means square root and everything inside the curly brackets {} are inside the square root, then it would be:
    -(-264) + or - /{(-264)^2 - 4*12*1089}
    everything divided by (2*12)

    264 + or - /{17424} all divided by 24

    264 + or - 132 all divided by 24

    now
    264 + 132 divided by 24 gives you 16.5
    and
    264 - 132 divided by 24 gives you 5.5
    therefore x = 5.5 and 16.5
    now we take the second derivative to find the maximum value
    so
    V' = (12X^2 - 264X + 1089)
    take the 2nd derivative of the equation above
    V'' = (24X - 264)
    now we plug in the 2 values of 'X' we got in this equation

    (24*16.5) - 264 = 132
    &
    (24*5.5) - 264 = -132

    since we want to maximize volume we would take 5.5 as the value of X because the answer is negative (the graph would be concave down, therefore maximum point)

    we know have the value of 'X' so plug it into the dimensions that we started with

    (33 - (2*5.5)) = 22
    therefore
    the dimensions that the box should be cut are 22 inches by 22 inches by 5.5 inches

    *** if you would like to know the maximum volume of the box plug in 5.5 into the volume equation

    V = (4X^3 - 132X^2 + 1089X)
    = (4*5.5)^3 - (132*5.5)^2 + (1089*5.5)
    = 2662 inches^3

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. CALCULUS

    A PIECE OF SHEET METAL IS 2.6 TIMES AS LONGS AS IT IS WIDE. IT IS TO BE MADE INTO A BOX WITH AN OPEN TOP BY CUTTING 3-INCH SQUARES FROM EACH CORNER AND FOLDING UP THE SIDES. IF THE VOLUME OF THE BOX MUST BE BETWEEN 600 AND 800 CUBIC …
  2. algebra

    Open-top box. Thomas is going to make an open-top box by cutting equal squares from the four corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides. If the area of the base is to be 80 square inches, then what …
  3. math

    open top rectangular box made from 35 x 35 inch piece of sheet metal by cutting out equal size squares from the corners and folding up the sides. what size squares should be removed to produce box with maximum volume.
  4. algebra

    rectangular open-topped box is made from a 9 x 16 piece of cardboard by cutting x-inch squares out of each corner and folding up the sides. What size square should be cut out to produce a volume of 120 cubic inches?
  5. Math

    An open box is to be made from a 10-ft by 14-ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and folding up the sides. what size squares should be cut to obtain a box with largest possible …
  6. Calculus

    An open top box is made by cutting congruent squares from the corners of a 12 inch by 9 inch sheet of cardboard and then folding the sides up to create the box. What are the dimensions of the box which contains the largest volume?
  7. inermediate algebra

    A box with no top is to be made by cutting a 2-inch square from each corner of a square sheet of metal. After bending up the sides, the volume of the box is to be 220 cubic inches. Find The the length of a side of the square sheet …
  8. pre-calc

    A box with an open top is to be constructed by cutting equal-sized squares out of the corners of a 18 inch by 30 inch piece of cardboard and folding up the sides. a) Let w be the length of the sides of the cut out squares. Determine …
  9. Math

    A box with a rectangular base and no top is to be made from a 9 inch by 12 inch piece of cardboard by cutting squares out of the corners and folding up the sides. What size (side-length) squares should be cut out to make the box have …
  10. Pre-algebra

    Hey I'm having a lot of trouble with this question. An open box is to be made from a 20 inch by 40 inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. A) What size should the …

More Similar Questions