check the picture below.
thus
[tex]\bf 32^2=w^2+\left( \cfrac{9w}{16} \right)^2\implies 32^2=w^2+\cfrac{(9w)^2}{16^2}\implies
32^2=w^2+\cfrac{9^2w^2}{16^2}
\\\\\\
32^2=\cfrac{16^2w^2+81w^2}{16^2}\implies 32^2\cdot 16^2=337w^2
\\\\\\
(32\cdot 16)^2=337w^2\implies
\sqrt{(32\cdot 16)^2}=\sqrt{337w^2}
\\\\\\
32\cdot 16=w\sqrt{337}\implies
\cfrac{32\cdot 16}{\sqrt{337}}=w\implies
\boxed{\cfrac{512}{\sqrt{337}}=w}\\\\
-------------------------------\\\\[/tex]
and we can rationalize that to [tex]\bf \cfrac{512}{\sqrt{337}}\cdot \cfrac{\sqrt{337}}{\sqrt{337}}\implies \boxed{\cfrac{512\sqrt{337}}{337}}[/tex]
thus, its height is [tex]\bf h=\cfrac{9w}{16}\implies h=\cfrac{9}{16}\cdot \cfrac{512\sqrt{337}}{337}\implies \boxed{h=\cfrac{4608\sqrt{337}}{5392}}[/tex]
