Its Young's (excluded) not Old's (included)

Young

We take an elemental disk at a distance $x$ from the bottom of thickness $dx$ .

Radius of disk is given by $r=\dfrac { xR }{ L }$ .

So weight of the cone below the disk $=mg\dfrac { \dfrac { 1 }{ 3 } \pi (\dfrac { { x }^{ 2 }R^{ 2 } }{ { L }^{ 2 } } )x }{ \dfrac { 1 }{ 3 } \pi { R }^{ 2 }L } =\dfrac { mg{ x }^{ 3 } }{ { L }^{ 3 } }$

Strain= $\dfrac { Weight }{ AY }=\dfrac { mgx }{ Y\pi { R }^{ 2 }L }$ , $A=Area \quad of \quad disk=\dfrac { \pi { x }^{ 2 }{ R }^{ 2 } }{ { L }^{ 2 } }$

Let the total elastic potential energy be $E$

$dE=\dfrac { 1 }{ 2 } Y{ (Strain) }^{ 2 }dV\quad V=volume$

$dV=\dfrac { \pi { x }^{ 2 }{ R }^{ 2 } }{ { L }^{ 2 } } dx$

Putting the values we have $dE=\dfrac { { m }^{ 2 }{ g }^{ 2 } }{ 2\pi Y{ R }^{ 2 }{ L }^{ 4 } } { x }^{ 4 }dx$

$E=\dfrac { { m }^{ 2 }{ g }^{ 2 } }{ 2\pi Y{ R }^{ 2 }{ L }^{ 4 } } \int _{ 0 }^{ l }{ { x }^{ 4 }dx }=\dfrac { { m }^{ 2 }{ g }^{ 2 }L }{ 10\pi { R }^{ 2 }Y }$

So we have $a=2,b=2,c=1,d=10,e=1,f=2$

Hence $\boxed { a+b+c+d+e+f=18 }$