It is given that,

Height of the cylindrical bucket $=32cm$

Radius of the cylindrical bucket $=18cm$

Height of conical heap $=24cm$

As we know that,

Volume of cylinder $=\pi \times {{r}^{2}}\times h$

And, volume of cone $=1/3\pi \times {{r}^{2}}\times h$

So, from the question

Volume of the conical heap $=$ Volume of the cylindrical bucket

$1/3\pi \times {{r}^{2}}\times 24=\pi \times {{18}^{2}}\times 32$

${{r}^{2}}={{18}^{2}}\times 4$

$r=18\times 2=36cm$

Then,

Slant height of the conical heap (l) is given by

$l=\sqrt{\left( {{h}^{2}}+{{r}^{2}} \right)}$

$l=\sqrt{\left( {{24}^{2}}+{{36}^{2}} \right)}=\sqrt{1872}$

$l=43.26cm$

Therefore, the radius and slant height of the conical heap are $36cm$ and $43.26cm$ respectively.