1. Questions

Calculus

A box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides. Find the value of h that maximizes the volume of the box if
A = 7 and B = 12

  1. 👍
  2. 👎
  3. 👁
Jackie

  1. Draw a rectangle 7 x 12.

    In the corners draw four squares of dimensions h x h.

    It doesn't matter what the length of h is.

    Dimensions of a new rectangle will be ( 7 - 2 h ) x ( 12 - 2 h )

    The volume of the box will be:

    Area of a new rectangle ∙ length h

    V = ( 7 - 2 h ) ∙ ( 12 - h ) ∙ h

    V = ( 84 - 24 h - 7 h + 2 h² ) ∙ h

    V = ( 2 h² - 31 h + 84 ) ∙ h

    V = 2 h³ - 31 h² + 84 h

    The function has an exstrem (maximum or minimum) where the first derivative is zero.

    In this case you need to find:

    dV / dh = V′( h ) = 0

    ( 2 h³ - 31 h² + 84 h )′ = 2 ∙ 3 h² - 31 ∙ 2 h + 84 = 0

    6 h² - 62 h + 84 = 0

    The solutios are:

    h = ( 31 -√457 ) / 6 ≈ 1.60374

    and

    h = ( 31 +√457 ) / 6 ≈ 8.7296

    Now you have to second derivative test.

    If V"( h ) < 0 then function has maximum at h.

    If V" ( h ) > 0 then function has minimum at h.

    V" ( h ) = 2 ∙ 6 h - 62

    V" ( h ) = 12 h - 62

    For

    h = 1.60374

    V" ( h ) = 12 ∙ 1.60374 - 62 = − 42​.75512 < 0

    For

    h = 8.7296

    V" ( h ) = 12 ∙ 8.7296 - 62 = 42.7552 > 0

    So

    For h = ( 31 -√457 ) / 6 ≈ 1.60374 volume has a maximum.

    For h = ( 31 +√457 ) / 6 ≈ 8.7296 volume has a minimum.

    By the way dimension h = 8.7296 are impossible because for h = 8.7296 dimension 7 - 2 h would have a negative value and the length cannot be negative.

    1. 👍
    2. 👎
    Bosnian

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus

    An open box is formed from a piece of cardboard 12 inches square by cutting equal squares out of the corners and turning up the sides, find the dimensions of the largest box that can be made in this way.

  2. 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

  3. 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

  4. 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

  1. math

    A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides, as shown in the figure. The box is to hold 324 in3. How big a piece of

  2. math

    an open box is to be formed out of a rectangular piece of cardboard whose length is 8 cm longer than its width to form the box,a square of side 4 cm will be removed from each corner of the cardboard then the edges of the remaining

  3. 11th grade math

    A box without a lid is constructed from a 38 inch by 38 inch piece of cardboard by cutting in. squares from each corner and folding up the sides. a) Determine the volume of the box as a function of the variable . b) Use a graphing

  4. 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

  1. 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

  2. Calculus

    You are given a piece of sheet metal that is twice as long as it is wide an has an area of 800m^2. Find the dimensions of the rectangular box that would contain a max volume if it were constructed from this piece of metal by

  3. Pre Calculus

    A piece of cardboard measuring 13 inches by 11 inches is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides. a. Find a formula for the volume of the box in terms of x b.

  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.