3. There are two cylinders with a volume ratio of $5: 8$. The lateral surfaces of these cylinders unfold into the same rectangle. If the length and width of this rectangle are both increased by 6, its area increases by 114. What is the area of this rectangle? $\qquad$ .
【Answer】 40
【Analysis】Let the length and width of the rectangle be $a$ and $b$; then the volumes of the two cylinders are: $\pi\left(\frac{a}{2 \pi}\right)^{2} b$ and $\pi\left(\frac{b}{2 \pi}\right)^{2} a$, respectively. Therefore, $\pi\left(\frac{a}{2 \pi}\right)^{2} b: \pi\left(\frac{b}{2 \pi}\right)^{2} a=a: b=8: 5$;

Also, according to the area increase of 114, we have: $(a+6)(b+6)-a b=114 \Rightarrow a+b=13$;
Solving the system $\left\{\begin{array}{l}a: b=8: 5 \\ a+b=13\end{array} \Rightarrow\left\{\begin{array}{l}a=8 \\ b=5\end{array} \Rightarrow 8 \times 5=40\right.\right.$.