Results 1 to 5 of 5

Math Help - Optimiztion=O

  1. #1
    Junior Member User Name's Avatar
    Joined
    May 2008
    Posts
    40

    Unhappy Optimiztion=O

    Hi to all! I've got quick but tricky question!
    see this and then answer the following question if you could=)

    x is depth of cone!
    how do we find volume of cone in terms of x? and maximum volume of cone?
    and value of x which maximizes the volume of cone?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jul 2009
    Posts
    397
    From the picture, you can find the height of the cone. For the base radius, you can use pythagoras. (all in terms of x)

    For maximum volume :
    \frac{dV}{dx} = 0 -----> solve for x
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member alunw's Avatar
    Joined
    May 2009
    Posts
    188

    Attempted solution

    Volume of right cone is 1/3*pi*r^2*h where r is radius of base and h is height.
    Let theta be the angle between the axis of the cone and a line from the centre of the sphere to a point on the base circle. Then we have:

    r=5*sin(theta)
    x =5*cos(theta)
    h = 5+x.

    So volume is 1/3*pi*25*sin^2(theta)*(5+5*cos(theta)), where clearly theta is somewhere between 0 and pi/2 radians.

    Put t = tan(0.5*theta) and use:

    sin(theta)=2t/(1+t^2)
    cos(theta)=(1-t^2)/(1+t^2)

    Now volume is 1/3*pi*25*4*t^2*(5*(1+t^2)+5*(1-t^2))/(1+t^2)^3 =
    = 1/3*pi*125*8*t^2/(1+t^2)^3

    Now let us put u=t^2
    Vol = 1/3*pi*1000*u/(1+u)^3

    So differentiating wrt u we see that the maximum is at a root of

    (1+u)^3-u*3*(1+u)^2=0
    (1+u)^2((1+u)-3u)=0
    u=-1 is impossible since u>=0
    so u=1/2
    but we had x = 5*cos(theta)=5*(1-t^2)/(1+t^2)=5*(1-u)/(1+u) = 5*0.5/1.5 = 5/3
    and the volume is 1/3*1000*pi*0.25/1.5^3
    Last edited by alunw; July 30th 2009 at 06:53 AM. Reason: Improved formatting
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member User Name's Avatar
    Joined
    May 2008
    Posts
    40
    many thanks
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Jul 2009
    Posts
    397
    You're welcome ^^
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Optimiztion problem involving a man on a boat
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 13th 2010, 07:38 PM

Search Tags


/mathhelpforum @mathhelpforum