Composite Bodies: Finding the Centroid

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Homework Statement

A single-scoop ice cream cone is a composite body made from a single scoop of ice cream placed into a cone. Assume that the scoop of ice cream is a sphere with radius r = 3.65 cm that is placed into a 9.90 cm tall cone. The interior height of the cone is 9.00 cm. The cone has an exterior radius of 3.10 cm and an interior radius of 2.80 cm. The scoop of ice cream sits on the cone's interior radius and extends into the cone some distance. Find the z-bar centroid for the cone (the scoop of ice cream and the cone).

Homework Equations

Components: Sphere (scoop of ice-cream) and Cone (ice-cream cone)

Volume of Sphere: V=4/3∏R³, V=4/3∏(3.65)³, V=203.6888249
Centroid of Sphere: 12.2
Product of Volume and Centroid: 12.2 x 203.6888249 = 2485.003664

Volume of Cone: V=1/3∏R²h, V=1/3∏(3.10)²(9.90), V=99.62932782
Centroid of Cone: 6.779
Product of Volume and Centroid: 99.62932782 x 6.779 = 675.3872133

The Attempt at a Solution

So, z-bar = 10.40 cm (Sum of product of volume and centroid for sphere and cone/Sum of volume of sphere and cone)

I tried that answer however, it was wrong.
I then realized that the cone is not a completely solid shape, with it being hollow inside and that it has two different radii and height.
As this image shows:

So, I then found the product of volume and centroid for the inner cone: volume being 73.89025921 and centroid being 6. Then, I subtracted this from the other product of volume and centroid for the cone and then found z-bar again. This was also wrong.

I'm not sure with what else there is to do. Any help, please? I've used all the hints for the question and I only have one attempt remaining.

 

[TSny]

View attachment 155785
So, I then found the product of volume and centroid for the inner cone: volume being 73.89025921 and centroid being 6. Then, I subtracted this from the other product of volume and centroid for the cone and then found z-bar again.

This is the right idea. But, the centroid for the inner cone (considered solid) is not located at z = 6. It is located at z = 0.90 cm +(3/4)(9.00 cm) = 7.65 cm