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Designing a 5L Football for the MFL

  1. Jul 20, 2016 #1
    1. The problem statement, all variables and given/known data
    You have been employed but(sic) the Mathematics Football League (MFL) to design a football. Using the volume of revolution technique, your football design must have a capacity of 5L ± 100mL. You must present a statement considering the brief below. Just a quick side note, I have checked and it cannot be spherical, it must be in the normal elliptical rugby or gridiron shaped ball (roughly).
    Brief:
    • The volume of revolution technique is to be used
    • The football must have a capacity of 5L ± 100mL
    • You should use a single non-linear function
    • You must explain carefully all the steps that you take in choosing the function and the dimensions of the football
    • You may use numerical methods (trapezium rule, numerical integration or a graphing package) in the design of the football
    2. Relevant equations
    1. V=π∫[f(x)]^2 dx (from a to b) -> Sorry I don't know how to add the boundaries in properly
    2. V=π∫[f(x)-g(x)]^2 dx (from a to b)
    3. V=π∫[f(y)]^2 (from a to b - along the y-axis)
    4. x(turning point)=-b/2a
    5. T=2π/b
    6. Quadratic Formula
    7. y=ax^2+bx+c
    8. y=sin(x)
    9. etc.
    3. The attempt at a solution
    Here's some of my attempts at the solution. (I have done it in a million different ways and cannot seem to get it, I either got somewhere between 2000 and 4500mL and 6000-7000mL. Most of my working is just me picking up small errors in the calculations. The processes in the photos are pretty much the different processes I attempted the solution with) Sorry for the bad camera, thanks in advance.
    nT9T3uT.jpg
    1aW4aNV.jpg
    gRZinCQ.jpg
    ZVFSGew.jpg
     
  2. jcsd
  3. Jul 20, 2016 #2

    berkeman

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    Staff: Mentor

    Welcome to the PF.

    Your uploaded pictures are pretty much unreadable. Can you scan them instead and upload the PDF images?
     
  4. Jul 21, 2016 #3

    RUber

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    Homework Helper

    I would recommend starting with the goal in mind.
    You want the volume to be 5L, and you need to have two symmetric zeros for your function.
    ## V = \pi \int_0^{x_1} [f(x)]^2 dx =5##
    I think that starting with a quadratic function is fine. Set the first zero at x = 0, so you have ##f(x) = ax^2 + bx## which as a second zero at x = -b/a.
    Square that to get your integrated function.
    ## V = \pi \int_0^{-b/a} ( ax^2 + bx )^2 dx = 5. ##
    The integration is straightforward, and you should be left with an equation with lots of a's and b's and a 5.
    You can pick a value for a and solve for b. Graph the resulting function (ax^2 + bx) so you can see what you get.
    Increasing a will decrease the length of your football and make it fatter in the middle. Decreasing a will make it longer and make it thinner in the middle.
    Of course, you need to make sure your units match up...so in this case, since 1L = (10cm)^3, your units of length for x are in decimeters.
     
  5. Jul 21, 2016 #4

    RUber

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    Homework Helper

    You could also consider entering "volume of an ellipsoid" into Google if you want another perspective.
     
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