Rate of radiation coming from inside a container

[cyclonefb3]

Homework Statement

A aluminum container has walls that are 23 mm thick. A radiation detector measures a rate of 542 Hz outside the container. The radiation source inside the container has a half-life of 2.4 years. What is the rate of radiation from the source inside the container?

Homework Equations

x_1/2=ln2/u

u= 0.014 mm^-1
x_1/2=48.26 mm

The Attempt at a Solution

I tried to solve this problem using proportions, and it didn't work. Is there another way?

23 mm is 47% of aluminums half-length (48.36 mm). so the amount of radiation change from inside to outside should be 50% of 47% of aluminums half-lenght.

so 542*23.5% = 127.3

542+127.3 = 669.3 Hz

 

[Irid]

Proportion doesn't work here. You should use the exponential law of decrease in intension:

$$I = I_0 2^{l/L}$$

where $$l$$ is the length of absorbing material and $$L$$ is the half-lenght.

 

[baba89]

Unfortunately, using proportions will not work in this situation. To solve this problem, we need to use the equation for radioactive decay: N=N0e^(-λt), where N is the final amount of radiation, N0 is the initial amount of radiation, λ is the decay constant, and t is the time elapsed.

In this case, we know that N0=542 Hz, t=2.4 years, and we need to solve for N. We also know that the decay constant, λ, is related to the half-life by the equation λ=ln2/t1/2.

Substituting the given values, we get λ=ln2/2.4 years=0.2887 years^-1.

Now, we can plug in all the values into the equation for radioactive decay:

N=N0e^(-λt)

N=542 Hz * e^(-0.2887 years^-1 * 2.4 years)

N=542 Hz * e^(-0.6929)

N=542 Hz * 0.5002

N=271.1 Hz

Therefore, the rate of radiation from the source inside the container is approximately 271.1 Hz.