How Small Must a Bone Be to Detect a 5% Change in X-ray Imaging Intensity?

[jdemps]

Homework Statement

Imagine a 10 cm thick slab of flesh. If your x-ray imager can reliably identify a 5% change in intensity from one location to another in an image, what is the smallest diameter bone you are going to be able to make detect in your x-ray image?

Absorption length of bone: 0.017 m

Homework Equations

I=I₀e^-μx

The Attempt at a Solution

I tried letting 0.05=e^-μx letting 0.017m be μ. The correct answer is supposed to be 1.3 mm but I can't seem to get that.

 

[BvU]

1. going from 1 to 0.05 is a change of 0.95. Consider going from I = 1 to I = 0.95.
2. μx must have no dimension, otherwise you can't exponentiate it. your exponent has the wrong dimension...
3. What is the definition of absorption length in your context ? I = I₀ / e or I = I₀ / 2 ?

 

[jdemps]

I set it up so that 0.95=e^-μx trying to let μ=1/0.017 so that x could be in meters without having units in the exponent. That did not work either when trying to solve for x. There are no explanations for these practice problems and I have only ever heard the term "absorption coefficient" not absorption length, which is why I'm confused.

 

[BvU]

"Did not work either" means you found 0.017 m * ln(0.95) = 0.00087 m ?

I agree the most common maeening for absorption length is length over which I = I₀ / e.

It's just because your 'right' answer corresponds to 0.017 m * ²log(0.95) that I asked for this context.

I can't think of anything else to help you with at this moment (bedtime+3h)...