A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm3 of water.

We have,

Radius of the upper end, $R=20 \mathrm{~cm}$ and

Radius of the lower end, $r=12 \mathrm{~cm}$

Let the height of the bucket be $h$.

As,

Volume of the bucket $=12308.8 \mathrm{~cm}^{3}$

$\Rightarrow \frac{1}{3} \pi h\left(R^{2}+r^{2}+R r\right)=12308.8$

$\Rightarrow \frac{1}{3} \times 3.14 \times h \times\left(20^{2}+12^{2}+20 \times 12\right)=12308.8$

$\Rightarrow \frac{3.14 h}{3} \times(400+144+240)=12308.8$

$\Rightarrow \frac{3.14 h}{3} \times 784=12308.8$

$\Rightarrow h=\frac{12308.8 \times 3}{3.14 \times 784}$

$\therefore h=15 \mathrm{~cm}$

So, the height of the bucket is 15 cm.