1. The formula for the volume of a cone is:
   \[V = \frac { 1} { 3} \pi r^ 2 h\]
2. Given:
   • Diameter = 14 mm, so the radius \(r = \frac { 14} { 2} = 7\) mm.
   • Volume \(V = 564.15\) mm\(^ 3\).
3. Substitute the known values into the volume formula:
   \[564.15 = \frac { 1} { 3} \pi ( 7) ^ 2 h\]
4. Simplify the equation:
   \[564.15 = \frac { 1} { 3} \pi ( 49) h\]
   \[564.15 = \frac { 49} { 3} \pi h\]
   \[564.15 = 16.3333 \pi h\]
5. Solve for \(h\):
   \[h = \frac { 564.15} { 16.3333 \pi } \]
   \[h = \frac { 564.15} { 16.3333 \times 3.14} \]
   \[h = \frac { 564.15} { 51.3146} \]
   \[h \approx 10.99\]
6. Round to the nearest whole number:
   \[h \approx 11\]
   Therefore, the height of the cone is approximately 11 mm.

Supplemental Knowledge

The volume \(V\) of a cone is given by the formula:
\[V = \frac { 1} { 3} \pi r^ 2 h\]
where \(r\) is the radius of the base, \(h\) is the height, and \(\pi \) (pi) is approximately 3.14.

Knowledge in Action

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