Question

Two cones have the same volume but different heights the taller cone is four times taller then the shorter cone your friend concludes that the radius of the shorter cone must be four times longer than the radius of the taller cone what is his error ?

Answer 1

The radius is part of the base, therefore the height being four times taller does not pertain to the radius. The radius would actually be smaller than the other cone.

Answer 2

He didn't factor in the square of the radius.

Let's plug in some numbers.

Let the height of cone #1 = 4
Let the radius of cone #1 = 2

Let the height of cone#2 =1
Solve for the radius of cone #2

Volume = 1/3 Bh
\[B=\pi r ^{2}\]
Since the volumes are equal to each other we have:
\[1/3(\pi)(2^{2})(4)=1/3(\pi)(r ^{2)}(1)\]\[16(\pi)/3=(\pi)r ^{2}/3\]\[16=r ^{2}\]\[4=r\]

So radius of cone #2 is only 2 times longer than cone #1