Calculus

A rectangular piece of tin has an area of 1334 square inches. A square tab of 3 inches is cut from
each corner, and the ends and sides are turned up to make an open box. If the volume of the box is
2760 cubic inches, what were the original dimensions of the rectangular piece of tin? Show the work
that leads to the answer.
2. Now, let’s maximize the volume of the box that can be made using the original dimensions you found
in problem #1. Carefully complete the sequence below.
a) Find the algebraic representation for the volume of the box, V(x), where x is the length of the square tab
in each corner. Express V(x) as a polynomial in factored form and standard form. Make a table of values
for x and V if the side length of the square tab is x = 1, 3, 5, 7, 9, 11, and 13 inches. Show the work that
leads to each answer. How does the volume of the box change when the size of the square tabs increases
in size?
b) Length, width, and height must be positive. Use this fact to find the domain of the volume function V(x)
in the form a < x < b. Explain. Justify your explanation by solving inequalities.
c) Find the algebraic representation for the area of the bottom of the box, A(x). Express A(x) as a
polynomial in factored form and standard form. Make a table of values for x and A if the side length of
the square tab is x =1, 3, 5, 7, 9, 11, and 13 inches. Show the work that leads to each answer. How does
the area of the bottom of the box change when the size of the square tabs increases in size?
d) Use a graphing calculator to sketch the graph of y = V(x). Use the viewing window [0, 24, 4] by
[-2500, 4000, 500]. Then use the 2
nd
– TRACE – maximum feature to calculate its maximum point.
What are the dimensions of the box, (l  w  h), with the maximum volume? What is the maximum
volume of the box? Draw and label a graph to support this answer. Label the coordinates of the x-and yintercepts
and relative maximum and minimum points. Values must be accurate to three decimal places.
e) Find and simplify the difference quotient of y = V(x). That is, find ௏(௫ା௛)ି௏(௫)

. (This will get a little
messy.) Then, substitute h with 0 and simplify. The expression you obtained is called the derivative of
y = V(x). (You will study derivatives in calculus.) Now, find the zeros of the derivative accurate to three
decimal places. What do you notice?

  1. 👍
  2. 👎
  3. 👁
  1. Wow, nice problem with very clear and precise instructions how to do it.
    I suggest you just follow the steps they ask you to perform.

    I will start you off.

    Let the finished box be x inches long, y inches wide and 3 inches high

    given: 3xy = 2760
    xy = 920

    original piece is (x+6) by (y+6)
    (x+6)(y+6) = 1334
    xy + 6x + 6y + 36 = 1334
    920 + 6(x+y) = 1298
    6(x+y) = 378
    x+y = 63 or y = 63-x

    sub into xy= 920
    x(63-x) = 920
    63x - x^2 - 920 = 0
    x^2 - 63x + 920 = 0
    (x-40)(x-23) = 0

    x = 40 inches or x = 23 inches
    If x=40, then y = 23
    If x = 23, then y = 40

    so the rectangle was 46 by 29 inches

    2. now let the tab to be cut out be x inches
    (this x is not the same as the x we just used)

    etc. carry on

    1. 👍
    2. 👎
  2. Thanks for the help, but do you understand what Im suppoosed to graph on 2D d) Use a graphing calculator to sketch the graph of y = V(x). Use the viewing window [0, 24, 4] by
    [-2500, 4000, 500]. Then use the 2
    nd
    – TRACE – maximum feature to calculate its maximum point.
    What are the dimensions of the box, (l  w  h), with the maximum volume? What is the maximum

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    1. Rosa, Roberto, Andrea, and Inno find an estimate for square root 10. Who has proposed the best solution? (1 point) Rosa: "Use square root 9 and square root 25 to estimate." Roberto: "I will use square root 4 and square root 9."

  2. Algebra 2

    Can you double check my answer The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 2,560 square inches. What are the dimensions of the window? (1 point) 50.6 inches by 50.6 inches 40

  3. Pre-Algebra

    8. Mark needs to cut a piece of glass to replace a broken window. He has four pieces of glass: 6 feet long, 5 feet long, 3 feet long, and 7 feet long. If the length of the glass he needs to cut is square root 20 feet long, which

  4. math

    A manufacturer of open tin boxes wishes to make use of pieces of tin with dimensions 8 in. by 15 in. by cutting equal squares from the four corners and turning up the sides. a. Let x inches be the length of the side pf the square

  1. math

    An open box is to be made from a rectangular piece of tin 12 inches long and 10 inches wide by cutting pieces of x-inch square from each corner and bonding up the sides. find the formula that expresses the volume of the box as a

  2. 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

  3. Math

    1. If a number is not a rational number, then it is _____. (1 point) an integer an irrational number a whole number a radical 2. A picture frame is 12 inches long and 9 inches wide. In inches, what is the diagonal length of the

  4. MATH

    An open box with a square base is to be made from a square piece of cardboard 24 inches on a side by cutting out a square of side x inches from each corner and turning up the sides.Graph V=V(x)

  1. Math

    a rectangular piece of cardboard is twice as long as it is wide . from each of its for corners, a square piece 3 inches on a side cut out. the flaps at each corner are then turned up to form an open box. if the volume of the box

  2. Math

    Consider the circles shown. Circle A has a radius of 6 inches. Circle B has a radius 20% greater than Circle A. The figure shows two circles labeled as Upper A and Upper B of different radii. The diameter of the circle Upper A is

  3. Math

    The area of a rectangular prism's base is 25 square inches and its height is 10 inches. Find the volume of this rectangular prism.

  4. Math

    1. If a number is not a rational number, then it is _____. (1 point) an integer an irrational number a whole number a radical 2. A picture frame is 12 inches long and 9 inches wide. In inches, what is the diagonal length of the

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