Question

Two cylinders have equal volume.Their heights are in the ratio 1:2.Find ratio of their radii? (9 std maths)

Answer 1

Solve for one height value equaling a multiple of the other, then substitute it into the formula for a cylinder's volume and use algebra to obtain

`r_1/r_2 = sqrt(2)`.

Explanation:

We have that the formula for a cylinder's volume is

`V_("cylinder") = pi r^2 h`

Where `r` is the radius and `h` is the height. The given height ratio is

`h_1/h_2 = 1/2`

Where `h_1` represents the height of the first cylinder, and `h_2` represents that of second cylinder. We could solve for

`h_1 = (h_2)/2`

or

`h_2 = 2 h_1`

both of which we can use, if we were to consider the ratio between their volumes being equal:

`(pi (r_1)^2 h_1)/(pi (r_2)^2 h_2) = 1`

We could multiply the volume of the second cylinder to both sides to get

`pi (r_1)^2 h_1 = pi (r_2)^2 h_2`

Now, let's see, what can we do? We can substitute `h_2 = 2 h_1`:

`pi (r_1)^2 h_1 = pi (r_2)^2 2 h_1`

It seems that `pi` and `h_1` both cancel out:

`(r_1)^2 = 2(r_2)^2`

Let's divide by `(r_2)^2`:

`((r_1)^2)/((r_2)^2) = 2`

And take the square root:

`r_1/r_2 = sqrt(2)`

We have just solved for the ratio between the radii, `sqrt(2)`.