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The correct answer is $6$. Applying the distributive property to the expression $r{y}^4\left(15y-9\right)$ yields $15r{y}^5-9r{y}^4$. Since $90y^5-54y^4$ is equivalent to $r{y}^4\left(15y-9\right)$, it follows that $90y^5-54y^4$ is also equivalent to $15r{y}^5-9r{y}^4$. Since these expressions are equivalent, it follows that corresponding coefficients are equivalent. Therefore, $90=15r$ and $-54=-9r$. Solving either of these equations for $r$ will yield the value of $r$. Dividing both sides of $90=15r$ by $15$ yields $6=r$. Therefore, the value of $r$ is $6$.