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)...

 

[HalfLostAndSearching]

I would approach this problem by first understanding the concept of x-ray imaging intensity and how it relates to the absorption of materials. X-ray imaging relies on the differential absorption of x-rays by different materials, with denser materials such as bones appearing more white in the image due to their higher absorption of x-rays.

To find the smallest diameter bone that can be detected in the x-ray image, we need to consider the absorption length of bone, which is given as 0.017 m in the homework statement. This means that for every 0.017 m of bone, the intensity of the x-rays passing through it will decrease by a factor of e (the natural logarithm base).

Now, let's consider the 10 cm thick slab of flesh mentioned in the homework statement. This is equivalent to 0.1 m. If we divide this by the absorption length of bone (0.017 m), we get approximately 5.88. This means that the x-ray intensity will decrease by a factor of e 5.88 times as it passes through the 10 cm thick slab of flesh.

To detect a 5% change in intensity, we need to be able to distinguish between an intensity of e-5.88 and e-5.88*0.95. This gives us a difference of approximately 0.006. This means that the smallest diameter bone that can be detected in the x-ray image will be the one that causes a 5% change in intensity, which is approximately 1.3 mm.

In summary, the smallest diameter bone that can be detected in the x-ray image is approximately 1.3 mm, considering the absorption length of bone and the ability of the x-ray imager to detect a 5% change in intensity.