1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Optimization of kite

  1. Nov 23, 2008 #1
    My job is to maximize the area of the kite so that it will fly better, faster, higher. A kite frame is to be made from six pieces of wood. The four pieces that form its border have been cut the lengths. So one of the top border is 2 cm long and the bottom border is 4 cm. The total border length is 2+2+4+4=12 cm (a) Show that the area of the kite is given by the function A(x) = x*(sqrt16-x^2) + (sqrt4-x^2).
    (b) How long should each of the cross-pieces to be maximize the area of the kite?
    For question a, please see my work as follows:-
    I have express the top portion of the vertical cross piece as Y and the bottom cross piece as Z. The horizontal cross pieces expressed as 2x.

    So, Y^2=2^2-x^2 therefore Y=sqrt(4-x^2), and the for Z=sqrt(16-x^2)
    Therefore the A(x)=x[sqrt(16-x^2) + sqrt(4-x^2)]
    The derivative of the expression d/dx is x(sqrt(16-x^2)+sqrt(4-x^2)
    From this derivative tried to find x but I couldn't get.
    My last simplified was this=-x^2sqrt(4-x^2)+(16-x^2)*sqrt(4-x^2)+1(4-x^2)*sqrt(16-x^2)-x^2*sqrt(16-x^2)
    Once I know the value of x, I can solve the 2nd part (b) by my self.
    Last edited: Nov 23, 2008
  2. jcsd
  3. Nov 23, 2008 #2

    Thats what I get when I shove the last line into some software. So I am guessing that you have went wrong somewhere... Unfotunately its too late at night for me to think clearly. This is just a heads up.
  4. Nov 24, 2008 #3


    Staff: Mentor

    Your expression for dA/dx looks fine, but I think you went wrong when you tried to simplify it.
    [tex]dA/dx = -x^2 (16 - x^2)^{-1/2} + (16 - x^2)^{1/2} - x^2 (4 - x^2)^{-1/2} + (4 - x^2)^{1/2}[/tex]

    From the first two expressions, pull out a factor of (16 - x^2)^(-1/2). From the last two expressions, pull out a factor of (4 - x^2)^(-1/2). You should get
    [tex]dA/dx = (16 - x^2)^{-1/2}(-x^2 + 16 - x^2) + (4 - x^2)^{-1/2}(-x^2 + 4 - x^2)[/tex]
    [tex]= \frac{16 - 2x^2}{\sqrt{16 - x^2}} + \frac{4 - 2x^2}{\sqrt{4 - x^2}}[/tex]

    Set this last expression to zero and solve for x.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook