the hoop rotates through qan angle of 3/2 pie radian in 1 second how many revolutions does the hoop make in 1 minute

a revolution is a full go-around the circle, namely a 2π angle, as opposed to pie, but yours sounds more delicious.

so, we know it does 3/2 π in 1 second, how many π does it do in 60 seconds(1 minute) ?

[tex]\bf \begin{array}{ccll} radians&seconds\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ \frac{3\pi }{2}&1\\\\ r&60 \end{array}\implies \cfrac{\frac{3\pi }{2}}{r}=\cfrac{1}{60}\implies \cfrac{\frac{3\pi }{2}}{\frac{r}{1}}=\cfrac{1}{60}[/tex]

[tex]\bf \cfrac{3\pi }{2}\cdot \cfrac{1}{r}=\cfrac{1}{60}\implies \cfrac{3\pi }{2r}=\cfrac{1}{60}\implies \cfrac{3\pi \cdot 60}{2}=r\implies 90\pi =r\\\\ -------------------------------\\\\ \textit{how many times }2\pi \textit{ goes into }90\pi ?\qquad \cfrac{90\pi }{2\pi }\implies 45[/tex]