The equation of the circle given is,
[tex]x^2+y^2+36-15y=0[/tex]Move 36 to the right hand side
[tex]\begin{gathered} x^2+y^2-15y=-36+0 \\ x^2+y^2-15y=-36 \end{gathered}[/tex]Complete the square on y
[tex]\begin{gathered} x^2+(y^2-\frac{15}{2}y+(\frac{-15}{2})^2)=-36+(-\frac{15}{2})^2 \\ x^2+(y^2-\frac{15}{2}y+\frac{225}{4})=-36+\frac{225}{4} \\ x^2+(y-\frac{15}{2})^2=\frac{81}{4} \end{gathered}[/tex]The general form of the equation of a circle at the origin is,
[tex]x^2+y^2=r^2[/tex]Comparing the equation given and the general form of the equation
[tex]r^2=\frac{81}{4}[/tex]Simplify
[tex]\begin{gathered} r=\sqrt{\frac{81}{4}}=\frac{9}{2} \\ \therefore r=\frac{9}{2}\text{ or 4.5} \end{gathered}[/tex]Hence, the answer is
[tex]r=\frac{9}{2}\text{ or 4.5}[/tex]