Math

posted by .

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)

  • Math -

    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

  • Math -

    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

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Math - Geometry

    A cylindrical solid has a cylindrical circular hold drilled out of the center. Bascially, it's a circular cylinder with a hollow spot right down the middle. Find the surface area of the resulting solid. radius of larger circle: 2in …
  2. geometry/algebra

    I asked this question yesterday, and I'm trying to understand it since I got the wrong answer - please explain: You need to make a soda can with a volume of 29 cubic inches. Find the surface area. calculate the following: a) How much …
  3. math

    the volume of an cylindrical can with a radius r cm and height h cm is 128000 c^3, show that the surface area of the can is A=2(22/7)r^2 + 246000/r. Find the value for r to minimize the surface area. *i know what the quest. ask but …
  4. Math-Algebra 2

    The surface area of a golf ball is 8.92 in ^2. The surface area of a softball is 53.78 in ^2. Find the ratio of the volumes of a golf ball to a softball. Surface Area = 4π r^2 and Volume = 4/3π r^3 I'm not sure if I did this …
  5. Calculus

    *Optimization problem* I'm okay at some optimization problems, but this one has me stumped. You work for a company that manufactures circular cylindrical steel drums that can be used to transport various petroleum products. Your assignment …
  6. Calculus

    an open topped cylinder has a volume of 125 cubic inches. determine the radius of the pot that will minimize it's surface area. What I have so far... radius =r keight =h V=πr²h 125/πr²=h SA = πr² + 2πr(h) = πr² …
  7. math

    find the radius and height of a cylindrical soda can with a volume of 21in^3 that minimize the surface area.
  8. Math - Calculus I

    Optimization Problem: Find the dimensions of the right circular cylinder of greatest volume inscribed in a right circular cone of radius 10" and height 24"
  9. Math

    What is the surface area and volume of a composite figure made up of a hemisphere and a cone in terms of pi?
  10. Calculus

    A company needs to make a cylindrical can that can hold precisely 1.5 liters of liquid. If the entire can is to be made out of the same material, find the dimensions (radius and height) of the can that will minimize the cost. Round …

More Similar Questions