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?

  1. 12
asked by MEG
  1. All we have to consider is the area of the cross-section, since the length is a constant.

    let the sides to be turned up at both ends be x inches long
    let the remaining base be y inches
    2x + y = 12
    y = 12-2x

    area = xy = x(12-2x) = 12x - 2x^2
    d(area)/dx = 12 - 4x = 0 for a max of area
    4x = 12
    x = 3

    So the turn-ups should be 3 inches

    posted by Reiny
  2. Greatest capacity will be achieved when the area of cross section will be maximum.
    Let X be the base ,so (12-X)/2 will be the height.
    Area= Base x height= X(12-X)/2
    =(12X-X^2)/2
    Following the maximum theory, we differentiate the area w.r.t X and equate it to zero.
    d/dX of(Area)=(12-2X)/2 =0
    or 6-X=0
    or X=6 inches(base)
    so, height= (12-6)/2 = 3 inches

    posted by profkn

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus

    A long rectangle sheet of metal, 15 inches wide is to be made into a rain gutter by turning up equal amounts on each side so that they are perpendicular to the sheet. How many inches should be turned up in order to give the gutter
  2. Algebra

    A box is to be formed from a rectangular piece of sheet metal by cutting squares measuring 5 inches on a side and then folding the sides. The piece of sheet metal is twice as long as it is wide. If the volume of the box is to be
  3. math

    a closed rectangular tank 180cm height, 300cm long and 200cm wide is made of metal sheet which costs 850 naira per square meter. calculate the cost of metal sheet used in making the tank
  4. 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
  5. Calculus

    A carpenter wants to make an open-topped box out of a rectangular sheet of tin 24 inches wide and 45 inches long. The carpenter plans to cut congruent squares out of each corner of the sheet and then bend the edges of the sheet
  6. spc

    a long strip of sheet metal 12 inches wide is to be formed into an open gutter with a rectangular cross-section as shown. what is the maximum possible area of the cross-section?
  7. 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
  8. math

    A rectangular piece of sheet metal is 4 ft longer than it is wide. The area of the piece of sheet metal is 45 ft2. Find its length and width.
  9. algebra

    A rectangular hole 2 centimeters wide and x centimeters long is cut in a rectangular sheet of metal 5 centimeters wide and y centimeters long.
  10. Chemistry

    a sheet of metal is 97.4 mm wide and 59.4 mm long. if it weighs 9.788g and the density of the metal is 7.23g/cm^3, what is the thickness of the sheet(in mm) ?

More Similar Questions