A tower $AB$ leans towards west making an angle $\alpha$ with the vertical. The angular elevation of $B$, the top most point of the tower, is ${60}^{\mathrm{o}}$ as observed from a point $C$ due east of $A$ at a distance of $10 \mathrm{ft}$ from $A.$ If the angular elevation of $B$ from a point $D$ due east of $C$ at a distance of $20 \mathrm{ft}$ from $C$ is ${45}^{\mathrm{o}}$, then the value of $2\tan \alpha$ is equal to