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Solutions: 10

which reduces to

Equivalently,

The base area must be an integer multiple of π, so let r

so h is rational. The volume V must also be an integer multiple of π, and

Therefore, k - 9 must divide 162, and so r

The only equable solution where r is also rational is the all-integer solution h = 8 and r = 6, for which V = S = 96 π. That cone is a solid of revolution of an equable right triangle with base 6 and height 8, having perimeter and area equal to 24.

h r ^{2}r Lateral surface area/π V/π=S/π 12 18 4.243 54 72 15 15 3.873 60 75 9 27 5.196 54 81 8 36 6 6.000 60 96 24 12 3.464 84 96 33 11 3.317 110 121 7 63 7.937 84 147 20/3 90 9.487 110 200 60 10 3.162 190 200 19/3 171 13.077 190 361

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