pre-calc

construction of a rain gutter: a piece of rectangular sheet metal is 20 inches wide. it is to be made into a rain gutter by turning the edges to form parallel sides. let x represent the length of each of the parallel.

a) give the restrictions on x.
b) determine a function of A that gives the area of a cross section of the gutter.
c) for what values of x will A be a maximum (and thus maximize the amount of the water that the gutter will hold)?
d) what is this maximum area?
e) for what values of x will the area of a cross section be less than 40 square inches?

  1. 👍
  2. 👎
  3. 👁
  1. Look at a cross-section of the gutter.
    let the height be x
    then the base width = 20-2x , assuming the gutter is open at the top

    a) in 20-2x , 0 < x < 10
    b) Area = x(20-2x)
    c) Area = 20x - 2x^2
    = -2(x^2 - 10x + 25 - 25 )
    = -2( (x-5)^2 - 25)
    = -2(x-5)^2 + 50
    d) maximum area is 50 when x = 5

    e) x(20-2x) < 40
    -2x^2 + 20x < 40
    x^2 - 10x + 20 > 0
    consider x^2 - 10x + 20 = 0
    x = (10 ± √20)/2
    = 5 ± √5

    so the area is less than 40 for
    5-√5 < x < 5+√5

    1. 👍
    2. 👎
  2. thank you! that helped a lot

    1. 👍
    2. 👎
  3. how did you figure c?

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. College Algebra

    A rectanguler piece of metal is 5 inches longer than it is wide. Square with sides 1 inches longer are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 644 inches, what are

  2. Math

    The strength S of a rectangular beam varies jointly as its width w and the square of its thickness t. If a wood beam 3 inches wide and 5 inches thick supports 300 pounds, how much can a similar beam 3 inches wide and 8 inches

  3. 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

  4. MAT 150

    A rectangular piece of metal is 15 inches longer than it is white squares with size 3 inches longer cut from the four corners and the flaps are for the upward to form an open box if the volume of the box is 750 inches^3 What were

  1. calculus

    A long rectangular sheet of metal, 12 inches wide, is to be made into a rain gutter by turning up two sides so that they are perpendicular to the sheet. How many inches should be turned up to give the gutter its greatest capacity?

  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. calculus

    An open box is to be made from a 21 ft by 56 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have.

  4. Math

    A rectangular sheet of paper 8.5 inches by 11 inches is folded and then taped so that one pair of opposite vertices coincide. What is the number of inches in the perimeter of the resulting pentagon? Express your answer as a

  1. Calculus

    A gutter with trapezoidal cross-section is to be made from a long sheet of stainless steel that is 30 cm wide by turning up one-fourth of the width on each side. What width across the top will give the maximum cross sectional

  2. math

    A rectangular piece of sheet metal is 4 ft longer than it is wide. (See the illustration.) The area of the piece of sheet metal is 96 ft2. Find its length and width

  3. calculus

    an open rectangular box is to be made from a piece of cardboard 8 inches wide and 8 inches long by cutting a square from each corner and bending up the sides. a. express the volume of the box as a function of the size x cutout

  4. Math Wrod Problem

    1. A long strip of copper 8 inches wide is to be made into a rain gutter by turning up the sides to form a trough with a rectangular cross section. Find the dimentions of the cross-section if the carrying capacity of the trough is

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