一个组合几何体由一个圆柱和两个半球组成。 圆柱的半径为 $p$，长度为 $2p$，而半球的半径为 $p$。 该几何体的总表面积是半径为 $p$ 且母线长为 $l$ 的实心圆锥总表面积的六倍。 求用 $p$ 表示 $l$ 的表达式。

1. Find the total external surface area of the composite solid.

The two hemispheres form one complete sphere of radius $p$.

Curved surface area of cylinder:
$2\pi rh=2\pi(p)(2p)=4\pi p^2$

Surface area of two hemispheres:
$4\pi p^2$

Total surface area of composite solid:
$4\pi p^2+4\pi p^2=8\pi p^2$

2. Write the total surface area of the cone and form an equation.

Total surface area of cone:
$\pi r^2+\pi rl=\pi p^2+\pi pl$

Given that the composite solid's surface area is six times the cone's total surface area,
$8\pi p^2=6(\pi p^2+\pi pl)$

3. Solve the equation for $l$ in terms of $p$.

$8\pi p^2=6\pi p^2+6\pi pl$
$2\pi p^2=6\pi pl$
$l=\frac{2\pi p^2}{6\pi p}=\frac{p}{3}$