Percentage of Electrons Tunneling Through?

[Miley]

Homework Statement

A stream of electrons is of energy E is incident on a potential barrier of height U and thickness d. Even though U >> E, 5% of the electrons tunnel through the barrier. If the thickness of the barrier decrease to 0.86 d, what percentage of the electrons will tunnel through?

Homework Equations

P(d) = 0.05 = exp(-2αd)
P(d*a) = exp(-2αd*a) = exp(-2αd)^a = 0.05^a

Here a is a constant.

Convert to %.

The Attempt at a Solution

At first I tried "reasoning" my way through it (haha), thinking that if the thickness (d) decreased from an arbitrary 1 (100%) to a 0.4 (40%), then that's a 60% change in the thickness, meaning the 5% would also change by 60%, making that a 3%.

But clearly...I'm missing something right in front of me. I guess what's throwing me this time is the equation offered in the hint of: P(d*a) = exp(-2αd*a) = exp(-2αd)^a = 0.05^a. Where did the "a" come from, and where do plug in numbers to convert it to said percent?

 

[haruspex]

Where did the "a" come from, and where do plug in numbers to convert it to said percent?

The hint is true for any a. You need to choose a value for a which is relevant to the problem.