TRY THIS calculus?

Equation of the sphere is

r^2+(h/2)^2 = R^2, where R is the radius of the shot, and r is the radius and h is the height of the cylinder.

Volume of the cylinder

= f(r)

= pi r^2 h

= 2pi r^2 sqrt(R^2 - r^2)

f'(r) = 0 => [2r(R^2 - r^2) - r^3] = 0

Solve for r,

r = Rsqrt(2/3)

h = 2Rsqrt(1/3)

Check: r^2+(h/2)^2 = R^2

f(Rsqrt(2/3)) = pi r^2h = 4piR^3/(3sqrt(3)), the maximum volume

the maximum weight

= [4pi R^3/(3sqrt(3))]/[(4/3)piR^3](12)

= 12sqrt(1/3)

= 6.9282 kg

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To the answerer below me,

When diameter = height, you don't have the maximum cylinder.

Check: Your ratio = 0.52, and my ratio = 1/sqrt(3) = 0.58.

Since the shot weighs 12 kg, the cylinder weights 6.9282 kg. Here I used kg force unit for weight.

This is an interesting question. What it's basically asking is what is the largest cylinder you can cut from a sphere and what fraction of the spheres volume is that cylinder.

If you look at a circle the the largest 4 sided shape you can fit in there is a square. Rotate that around an axis and you'll get a cylinder. So the largest cylinder you could cut from a sphere would be one where the height equaled the diameter. This would correspond to a chord segment of theta = 90 degrees. Personally I find it easier to assign an arbitrary dimension to a sphere for purposes of calculation but you don't have to. Let's say our sphere has a diameter of 1m. From that our chord length would be (2)(0.5)(sin45) = 0.71m.

So our cylinder has dimensions of diameter = 0.71m and height = 0.71m.

The volume of the cylinder = (3.14)(0.35m^2)(0.71m) = 0.273m^3.

And the volume of the sphere would be (4/3)(3.14)(0.5m^3) = 0.523m^3

The ratio of volumes is then 0.273m^3/0.523m^3 = 0.52.

So the maximum volume cylinder you can make in a sphere would have a ratio of volume to the sphere of 0.52. The maximum mass would be equivalent or (0.52)(12kg) = 6.24kg.

Since the question asks for weight you multiply by gravity

(6.24kg)(9.8m/s^2) = 61.2N

the guy above me is correct, so theres no point me working it out. another way is to find the equation in a quadratic, find derivitave, let that = 0 and that is how you get max/min answers. but the question is asked poorly. it should be maximum MASS, not WEIGHT. weight invoves the external influence of gravity, which shouldnt be necessary