Solution:
Given:
[tex]\bullet \ \ \text{Volume of party hat} = 75\pi \text{ inches}^{3}[/tex]
[tex]\bullet \ \ \text{Radius of party hat:}\ 5 \ \text{inches}[/tex]
Recall that the formula to find the volume of a cone is πr²h/3. Now, let's use the formula to find the height.
[tex]\bullet \rightarrow \text{Volume of cone:} \ \dfrac{ \pi r^{2} h}{3}[/tex]
[tex]\bullet \rightarrow \text{Volume of party hat} =\dfrac{ \pi r^{2} h}{3} = 75\pi \text{ inches}^{3}[/tex]
[tex]\bullet \rightarrow \dfrac{ (\pi )( 5^{2})( h)}{3} = 75\pi \text{ inches}^{3}[/tex]
[tex]\bullet \rightarrow (\pi )( 5^{2})( h)} = 75\pi \times 3[/tex]
[tex]\bullet \rightarrow { (\pi )( 25)( h)}= 225\pi[/tex]
[tex]\bullet \rightarrow \dfrac{{ (\pi )( 25)( h)}}{25} = \dfrac{225\pi}{25}[/tex]
[tex]\bullet \rightarrow { (\pi )( h)}= 9\pi[/tex]
[tex]\bullet \rightarrow \dfrac{{ (\pi )( h)}}{\pi } =\dfrac{ 9\pi}{\pi }[/tex]
[tex]\bullet \rightarrow \boxed{\bold{h= 9 \ \text{inches}}}[/tex]
Thus, the height of the party hat is 9 inches.
