Results 1 to 2 of 2

Math Help - optimization

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    22

    optimization

    The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one 4 times as strong as the other, are placed 20 feet apart, where should an object be placed on the line between the sources so as to receive the least illumination?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,909
    Thanks
    771
    Hello, singh1030!

    I get an ugly cubic to solve . . .


    The illumination of an object by a light source is directly proportional to the strength
    of the source and inversely proportional to the square of the distance from the source.
    If two light sources, one 4 times as strong as the other, are placed 20 feet apart,
    where should an object be placed on the line between the sources
    so as to receive the least illumination?

    Let S = strength of the light source
    and d = distance fronm the light source.

    We are told that the illumination is: . I \:=\:\frac{kS}{d^2}
    Code:
         (S)                         (4S)
          * - - - - * - - - - - - - - *
          A    x    P      20-x       B

    Let A be the source with strength S
    and B be the source with strength 4S.

    The object is placed at P, which is x feet from A and (20-x) feet from B.

    The object's illumination from A is: . I_A \:=\:\frac{kS}{x^2}
    The object's illumination from B is: . I_B \:=\:\frac{k(4S)}{(20-x)^2}

    The total illumination is: . I \;=\;\frac{kS}{x^2} + \frac{4kS}{(20-x)^2} \;=\;kS\left[x^{-2} + 4(20-x)^{-2}\right]

    . . and that is the function we must minimize . . .

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. optimization help!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 12th 2009, 01:54 AM
  2. Optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 8th 2009, 03:09 PM
  3. Optimization
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 29th 2009, 11:56 AM
  4. optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 12th 2008, 11:47 AM
  5. Optimization
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: October 13th 2008, 07:44 PM

Search Tags


/mathhelpforum @mathhelpforum