Solving for energy involving hyperbolic

[nick227]

Homework Statement

What energy (in eV) should a beam of electrons have so that 0.1% of them are able to tunnel through a barrier of height 7.0eV and 1.0 nm wide? Start with the equation for T(E) and set it up with 1/T(E) on one side and let E/U=x for the unknown. Solve the equation for x and then E.

Homework Equations

T(E) (1+.25(U²/(E(U-E)))sinh²(_\alphaL))

\alpha = ((2m(U-E))^1/2)/h

The Attempt at a Solution

I get
0=x-x²-.25sinh²(_\alphaL)

this is after I make E/U=x. How can i solve for x and then solve for E?

 

[buffordboy23]

Have you tried to look for a numerical solution?

EDIT: If you need to solve it analytically, you may need to rewrite the hyperbolic sine term another way:

http://en.wikipedia.org/wiki/Hyperbolic_tangent#Standard_algebraic_expressions

I actually derived this whole equation once. It required the use of an alternate expression for the hyperbolic sine term.

The numerical solution seems easiest.