Mechanics 2 centre of mass question

I'm having trouble with this question.

A stone sculpture consists of a 2m x 1.6 m x 1 m cuboid surmounted by a cylinder of radius x metres and length 1 m (as shown in the diagram I can provide if you deem it necessary). The centre of mass of the sculpture is a point of contact between two parts. Find x.


so, I thought this was pretty easy, all I have to do is find x assuming the centre of mass is 1.6 metres upwards.

so I did: 1.6 = [ (1.6 x 2 x 0.8) + (pi x^2(x +3.2)) ] / (3.2 + pi x^2)

Which expands out to 2.56 = pi x^3 + 1.6 pi x^2.

I panicked and decided that this was wrong at this point, because x^3 is not something I would say looks like it is going to give me a good answer.

By the way, the actual answer is 0.934

Thanks in advance if you care to help out! (sorry about the lack of bbcode maths - I can't use it for the life of me)

where did x+3.2 come from?

Anyone see my mistake(s)?

Because the distance between the base and the midpoint of the cylinder is the height of the block + radius of the cylinder? I hope?

I'd like to see the diagram.

Your figures don't look consistent though. You say x+3.2 is the height of the centre of mass of the cylinder, implying the cuboid is 3.2 high? Which isn't consistent with it measuring 2 x 1 x 1.6

The midpoint of the cylinder will only be 0.5 above its base. i.e. half of its length. You shouldn't have x in there at all.

Yes, you're right! It definitely should be 1.6. EDIT: With the correct value, I have solved this much more easily (no cubic equation) Thanks so much - I can't believe I didn't spot that in all the tries I had :P
Here's the diagram.


but the midpoint of the cylinder from the base is the height of the block (1.6) + the radius of the cylinder - or am I talking rubbish?