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Question

The number of solid cones with integer radius and integer height each having its volume numerically equal to its total surface area is
  1. 2
  2. infinite
  3. 0
  4. 1


Solution

The correct option is D 1
Let radius be 'r' and height be 'h' of the solid cone, where r and h are integers
Volume = surface area
13πr2h=πrl+πr2    [l=r2+h2]
rh3r=3r2+h2
Squaring both sides,
r2h2+9r26r2h=9(r2+h2)r2h29h2=6r2hh2(r29)=6r2hh=6r2r29=6+54r29r2=9+54h6
since r and h are integer, so r2 will be perfect square,
for r2=16,hIr2=25,hIr2=36,h=8r2=49,hI
Hence there is only 1 possible integer values of r and h.

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