Designing a 5L Football for the MFL

[DPCalculus]

Homework Statement

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

Homework 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.

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.

 

[berkeman]

Welcome to the PF.

Your uploaded pictures are pretty much unreadable. Can you scan them instead and upload the PDF images?

 

[RUber]

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.

 

[RUber]

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