What is the number multiplied to the height of the cone to find the height of the cylinder?

Question

Both the cone and cylinder has the same volume and base length.

Answer 1

Use 2 formulas.

Volume of a cylinder: V = 3.14r^2 * h

Volume of a cone: V = 3.14r^2 * (h/3)

You may notice that the only difference between the 2 formulas is that h in the cone formula is replaced by (h/3) in the cylinder formula. This is an important clue!

Therefore it may be true that for any given cylinder, we can multiply the height by 3 to get the height of a cone with the same volume and radius. This makes sense when you picture a cone and a cylinder with the same radius in your mind. Clearly, the cone will have to be a lot taller in order to occupy the same amount of 3-d space.

Let's test this. Suppose a cylinder and a cone both have a volume of 20 and a radius of 2.

Let's solve the height for the cylinder.

20=3.14*2^2 * h

20=3.14*4h

20/(3.14*4)=h

5/3.14=h

1.592=h

Now let's solve the height of the cone.

20=3.14*2^2 (h/3)

20=3.14*4 (h/3)

20/(3.14*4) = h/3

5/3.14= h/3

15/3.14 = h

4.777=h

You will notice 4.777 is 3 * 1.592. Therefore the answer is one-third. (1/3)