Math

Optimization Problem

A right circular cylindrical can of volume 128tπ cm^3 is to be manufactured by a company to store their newest kind of soup. They want to minimize the surface area of the can to keep costs down.

What are the dimensions of the can with minimum surface area?

The volume of a cyllinder is V= πr^2h, where r is the radius and h is height. The surface area of a cylinder is SA= 2πr^2+2πrh, which is the sum of the area of the top and bottom (2 circles) and the area of the other curved sides (a rectangle, whose length is the circumference of the circles)

  1. 👍
  2. 👎
  3. 👁
  1. They have provided you with all the formulas you need
    given: πr^2h = 128
    h = 128/(πr^2)

    so in
    SA = 2πr^2+ 2πrh
    = 2πr^2+ 2πr(128/(πr^2)
    = 2πr^2 + 256/r
    d(SA)/dr = 4πr - 256/r^2
    = 0 for a min SA
    4πr = 256/r^2
    r^3 =64/π
    r = 4/π^(1/3) = appr 2.7311
    then h = 128/(πr^2) = 5.46223

    notice that this is twice the radius
    So the minimum SA is obtained when the
    radius is 2.4311 cm
    and the height is 5.46223 cm

    1. 👍
    2. 👎
  2. given: πr^2h = 128π
    h = 128π/(πr^2)

    TSA = 2πr^2+ 2πrh
    = 2πr^2+ 2πr(128/(r^2))

    d(tsa)/dr = 4πr - (256π/r^2)

    0 = 4πr - (256π/r^2)
    4πr = (256π/r^2)
    (solve for r)
    r^3 = 64
    r = 4

    (solve for h)
    h = 128/(4^2)
    = 8

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus

    I actually have two questions: 4. An open box is to be made from a rectangular piece of material 3m by 2m by cutting a congruent square from each corner and folding up the sides. What are the dimensions of the box of the largest

  2. math

    a right circular cylindrical container ,contains 2160cubic cm water. if the diameter of the base of the cylindrical container is 24 cm, then find the height of the cylindrical container.

  3. calculus2

    Use cylindrical shells to find the volume V of the solid. A sphere of radius r

  4. calculus

    1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln(5x), y = 1, y = 3, x = 0; about the y-axis 2. Use the method of cylindrical shells to find the volume V

  1. AP Calculus

    A cardboard box of 108in cubed volume with a square base and no top constructed. Find the minimum area of the cardboard needed. (Optimization)

  2. mathe

    Show that a right-circular cylinder of greatest volume that can be inscribed in a right-circular cone that has a volume that is 4/9 the volume of the cone.

  3. math

    In a study of the domestic market share of the three major automobile manufacturers A, B, and C in a certain country, it was found that their current market shares were 60%, 30%, and 10%, respectively. Furthermore, it was found

  4. calculus help

    A drinking cup is made in the shape of a right circular cylinder. for a fixed volume, we wish to make the total material used, the circular bottom and the cylindrical side, as small as possible. Find the ratio of the height to the

  1. Calc

    A 100 inch piece of wire is divided into 2 pieces and each piece is bent into a square. How should this be done in order of minimize the sum of the areas of the 2 squares? a) express the sum of the areas of the squares in terms of

  2. calculus2

    Use cylindrical shells to find the volume V of the solid. A right circular cone with height 9h and base radius 5r. The answer is 75πhr^2 but my answer is 39πhr^2. How???

  3. Maths

    The capacity of a cylindrical tank is 32l .It is 4/5 full of water .A quarter of this volume is poured into a pail .How much water is left in the cylindrical tank

  4. AP CALCULUS PLEASE HELP DUE TOMORROW!

    5. What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of .8L?

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