Question 62: Given positive real numbers $a$ and $b$ satisfy $a(a+b)=27$, try to find the maximum value of $a^{2} b$.

保留源文本的换行和格式，直接输出翻译结果。 

(注意：最后一句是解释说明，不是原文的一部分，因此在最终输出时应去掉。)

Question 62: Given positive real numbers $a$ and $b$ satisfy $a(a+b)=27$, try to find the maximum value of $a^{2} b$.
Question 62, According to the AM-GM inequality: $27=a(a+b)=a \cdot\left(a+\frac{b}{2}+\frac{b}{2}\right) \geq a \cdot 3 \sqrt[3]{a \cdot \frac{b}{2} \cdot \frac{b}{2}}$, rearranging gives $a^{2} b \leq 54$, equality holds when $a=3, b=6$!