Question

The height of a cylinder with constant volume is inversely proportional to the square of its radius. If h = 8 cm when r = 4 cm, what is r when h = 2 cm?

Answer 1

see the explanation..

Explanation:

`Height prop 1/(radius^2)`

This is what the above statement says about the inverse relationship between HEIGHT and SQUARE OF RADIUS.

Now in next step when removing the proportional sign `(prop)` we use an equal to sign and multiply `color(RED)"k"` on either of the sides like this;

`Height = k*1/(Radius^2)`

{where k is constant (of volume)}

Putting the values of height and radius^2 we get;

`8 = k*1/4^2`

`8 * 4^2= k`

`8 * 16= k`

`k= 128`

Now we have calculated our constant value `color(red)"k"` which is `color(red)"128"`.

Moving towards your question where radius is to be calculated.
Plugging the values into the equation:

`Height = k*1/(Radius^2)`

`2 = 128*1/r^2` {r is for radius}

`r^2=128/2`

`r^2=64`

`sqrt(r^2) =sqrt 64`

`r = 8`

Hence, for height of 2 cm with a constant of 128 we get the `color(blue)(radius)` of `color(blue)(2 cm)`