Sphere Inscribed AFL style mathematical football (or better known as sphere)

[loongsheng]

Hi guys, i'm in some trouble right now so anyway, this is what the question is.
Use the method of Lagrange multipliers to find the cylinder of maximum volume that can be inscribed inside an AFL style mathematical football with the geometric equation, f(x,y,z) = 15x^2 + 15y^2 + 4z^2 = 15.
without lose of generality, let the axis of the cylinder be the z axis.

a) Using r^2 = x^2 + y^2 , restate f(x,y,z) = 15 as g(r,z) = 15.
b) show that the volume of an inscribed cylinder , V(r,z) = 2 pi R^2 z.
c) Set-up and solve the Lagrange multiplier equations that maximises V(r,z) subject to g(r,z) = 15.
d) state the exact and approximate numberical values of (i) radius, (ii) height, (iii) volume of the inscribed cylinder of maximum volume.

SO,
does "let the axis of the cylinder be the z axis" mean that z would be the height?
if not,
I know my function to be maximised is V = pi r^2 h,
and my working for part a) is,
R^2 + (h/2)^2 - r^2 = 15 (R being the radius of the cylinder and r being the radius of the sphere)
i've managed to work it out by setting R^2 as the subject and plugging it back into my volume equation, but since this part i seem to be lost.
I also don't fully understand the question,
am I right to set my part (a) equation to 15?
cause if I am right, i can't seem to prove part (b) , and of course can't move on.

 

[HallsofIvy]

Hi guys, i'm in some trouble right now so anyway, this is what the question is.
Use the method of Lagrange multipliers to find the cylinder of maximum volume that can be inscribed inside an AFL style mathematical football with the geometric equation, f(x,y,z) = 15x^2 + 15y^2 + 4z^2 = 15.
without lose of generality, let the axis of the cylinder be the z axis.

a) Using r^2 = x^2 + y^2 , restate f(x,y,z) = 15 as g(r,z) = 15.
b) show that the volume of an inscribed cylinder , V(r,z) = 2 pi R^2 z.
c) Set-up and solve the Lagrange multiplier equations that maximises V(r,z) subject to g(r,z) = 15.
d) state the exact and approximate numberical values of (i) radius, (ii) height, (iii) volume of the inscribed cylinder of maximum volume.

SO,
does "let the axis of the cylinder be the z axis" mean that z would be the height?

No, of course not. z is the variable being measured along that axis. The ellipsoid intersect the z-axis at \(\displaystyle (0, 0,\sqrt{15}/2)\) and \(\displaystyle (0, 0, -\sqrt{15}/2)\).

if not,
I know my function to be maximised is V = pi r^2 h,
and my working for part a) is,
R^2 + (h/2)^2 - r^2 = 15 (R being the radius of the cylinder and r being the radius of the sphere)

What sphere???

i've managed to work it out by setting R^2 as the subject and plugging it back into my volume equation, but since this part i seem to be lost.
I also don't fully understand the question,
am I right to set my part (a) equation to 15?
cause if I am right, i can't seem to prove part (b) , and of course can't move on.

 

[loongsheng]

No, of course not. z is the variable being measured along that axis. The ellipsoid intersect the z-axis at \(\displaystyle (0, 0,\sqrt{15}/2)\) and \(\displaystyle (0, 0, -\sqrt{15}/2)\).


What sphere???

Sorry, what I meant by sphere was the AFL style mathematical football with geometric equation of 15x^2 + 15y^2 + 4z^2 = 15,
and could you please describe the question in lame-man terms? (sorry) and point me in the right direction?

 

[HallsofIvy]

The point is that, since there is NO sphere, there is no "R". Instead, look at the graph of 15x^2+ 4z^2= 15 in the xz-plane, since the ellipsoid is symmetric in y and z. That is an ellipse with axis, in the x direction, from (-1, 0, 0) to (1, 0, 0) and axis, in the z direction, from \(\displaystyle (\frac{2}{\sqrt{15}}, 0, 0)\). Now, inscribe a rectangle in that ellipse with height from (-h, 0, 0) to (h, 0, 0). What are the z values of the points on the ellipse with x= -h and h?