Calculus

Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

b) Suppose that in part a the original piece of cardboard is a square with side lengths k. Find the volume of the largest box that can be formed in this way.

c)Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length k and L. Holding k fixed, find the size of the corner squares x that maximizes the volume of the box as L→∞.

I solved part a, for x to be approximately .5657 and the volume at approximately 3.03. On part b and c, I am not even sure where to begin.

  1. 👍
  2. 👎
  3. 👁
  1. in inches,
    v = (36-2x)(48-2x)x
    dv/dx = 12(x^2-28x+144)
    dv/dx=0 when x = 6.789" ≈ 0.5657 ft

    For (b), just plug in k, and you have
    v = x(k-2x)^2
    dv/dx = 12x^2 - 8kx + k^2
    dv/dx=0 when x = k(2±√3)/6
    The max will occur at x = k(2-√3)/6, since (2+√3)/6 > 1/2, and the corners would use up the whole square.

    For (c), work the same way, but
    v = x(k-2x)(L-2x)

    1. 👍
    2. 👎
  2. thank you so much steve. great help

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    An open-topped box can be created by cutting congruent squares from each of the four corners of a piece of cardboard that has dimensions of 20cm by 30cm and folding up the sides. Determine the dimensions of the squares that must

  2. math

    An open box is made from a rectangular piece of cardboard measuring 16 cm by 10cm. Four equal squares are to be cut from each corner and flaps folded up. Find the length of the side of the square which makes the volume of the box

  3. math

    a 5cm by 5cm square is cut from each corner of a rectangular piece of cardboard.the sides are folded up to make an open box with a maximum volume.if the perimeter of the base is 50cm,what are the dimensions of the box.

  4. Calculus (Optimization)

    A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each corner and bending up the sides. What size square should be cut from each corner for the box to have

  1. math

    an open rectangular box is to be formed by cutting identical squares, each of side 2 in, one from each corner of a rectangular piece of cardboard, and then turning up the ends. If the area of the piece of cardboard is 160 in² and

  2. Math

    Nathan is designing a box to keep his pet newt in. To make the box, he’s going to start with a solid rectangle and cut squares with sides x cm in length from each corner, as shown in the picture below. The dimensions of the of

  3. Mathematics

    An open box is made from a square piece of sheet metal 19 inches on a side by cutting identical squares from the corners and turning up the sides. Express the volume of the box, V, as a function of the length of the side of the

  4. College Algebra

    A rectangular piece of metal is 15in longer than it is wide. Squares with sides 3 in long are cut from four corners and the flaps are folded up to form an open box. If the volume is 1218in^3, what we’re the original deminsions

  1. math

    A square piece of cardboard is to be used to form a box without a top by cutting off squares, 5cm on a side, from each corner and then folding up the sides. if the volume of the box must be 320 sq. sm, what must be the length of a

  2. Maths

    a rectangular metal sheet of length 30cm and breadth 25cm is to be made into an open box of base area 300cm by cutting out equal squares from each of the four corners and then bending up the edges find the length and the side cut

  3. Math

    A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in by 12 in by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the

  4. Math

    A rectangular piece of cardboard measuring 12 cm by 18 cm is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each square in

You can view more similar questions or ask a new question.