# 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?

Jul 10, 2016

see the explanation..

#### Explanation:

$H e i g h t \propto \frac{1}{r a \mathrm{di} u {s}^{2}}$

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 $\left(\propto\right)$ we use an equal to sign and multiply $\textcolor{R E D}{\text{k}}$ on either of the sides like this;

$H e i g h t = k \cdot \frac{1}{R a \mathrm{di} u {s}^{2}}$

{where k is constant (of volume)}

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

$8 = k \cdot \frac{1}{4} ^ 2$

$8 \cdot {4}^{2} = k$

$8 \cdot 16 = k$

$k = 128$

Now we have calculated our constant value $\textcolor{red}{\text{k}}$ which is $\textcolor{red}{\text{128}}$.

Plugging the values into the equation:

$H e i g h t = k \cdot \frac{1}{R a \mathrm{di} u {s}^{2}}$

$2 = 128 \cdot \frac{1}{r} ^ 2$ {r is for radius}

${r}^{2} = \frac{128}{2}$

${r}^{2} = 64$

$\sqrt{{r}^{2}} = \sqrt{64}$

$r = 8$

Hence, for height of 2 cm with a constant of 128 we get the $\textcolor{b l u e}{r a \mathrm{di} u s}$ of $\textcolor{b l u e}{2 c m}$