Question

m4-3.4 interpreting logarithmic models – collaborative problem
many models of household smoke detectors use a radioactive element as a component in the smoke-detection system. in one model, the radioactive element has a half-life of 450 years. this model will stop working (even if you change the battery annually) if the radioactivity drops to 0.2% of its original radioactivity level. the amount of radioactivity r(t) left after t years of radioactive decay is related to the original amount of radioactivity r₀ by the exponential function r(t) = r₀·e⁻ᶜᵗ, where c is a constant that is unique to the radioactive element.

task:

a) how long can this smoke detector be expected to function properly?

b) based on this result, explain why it is recommended that the battery in a smoke detector be changed annually.

Explanation:

Part (a)

Step1: Find the constant \( c \) using half - life

The half - life \( T = 450 \) years. At \( t = T\), \( R(T)=\frac{R_0}{2}\). Substitute into \( R(t)=R_0e^{-ct}\):
\(\frac{R_0}{2}=R_0e^{-c\times450}\)
Divide both sides by \( R_0 \) (since \( R_0
eq0\)): \(\frac{1}{2}=e^{- 450c}\)
Take the natural logarithm of both sides: \(\ln(\frac{1}{2})=-450c\)
We know that \(\ln(\frac{1}{2})=-\ln(2)\), so \(-\ln(2)=-450c\)
Solve for \( c \): \( c=\frac{\ln(2)}{450}\approx\frac{0.6931}{450}\approx0.00154\)

Step2: Find \( t \) when \( R(t) = 0.002R_0\)

Substitute \( R(t)=0.002R_0\) and \( c=\frac{\ln(2)}{450}\) into \( R(t)=R_0e^{-ct}\):
\(0.002R_0=R_0e^{-\frac{\ln(2)}{450}t}\)
Divide both sides by \( R_0 \): \(0.002 = e^{-\frac{\ln(2)}{450}t}\)
Take the natural logarithm of both sides: \(\ln(0.002)=-\frac{\ln(2)}{450}t\)
Solve for \( t \): \(t=-\frac{450\times\ln(0.002)}{\ln(2)}\)
Calculate \(\ln(0.002)\approx - 6.2146\) and \(\ln(2)\approx0.6931\)
\(t =-\frac{450\times(-6.2146)}{0.6931}=\frac{450\times6.2146}{0.6931}\approx\frac{2796.57}{0.6931}\approx4035\) years

The radioactive element in the smoke detector has a very long effective life (about 4035 years) before its radioactivity drops to a non - functional level. However, the battery in the smoke detector has a much shorter lifespan (usually a few years at most). Changing the battery annually ensures that the battery does not die, so that if there is a fire, the smoke detector (which is still functional due to the radioactive element) can detect the smoke and alert the occupants.

Answer:

The smoke detector can be expected to function properly for approximately \(\boldsymbol{4035}\) years.

Part (b)