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Question #43700

Difficulty: 8/10 | 🟢 Free

The Question

In a geological rock sample, the ratio of the number of stable daughter nuclei ( $N_d$ ) to the number of radioactive parent nuclei ( $N_p$ ) is measured to be $R = \frac{N_d}{N_p} = 15.0 \pm 0.48$ . It is assumed that the sample initially contained no daughter nuclei and that all daughter nuclei present are solely due to the decay of the parent nuclei. If the decay constant $\lambda$ of the parent nuclei is known exactly, what is the percentage error in the estimated age $t$ of the rock sample?

Options

A.
$\frac{3}{4 \ln 2} \%$
B.
3.2 %
C.
3.0 %
D.
$\frac{16}{5 \ln 15} \%$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Activity Units and Transitions|Relative Error Propagation in Powers|Significant Figures|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly propagate errors through non-linear (specifically logarithmic) functions in a physical context.

A common lethal trap is applying linear relative error propagation directly to the ratio ( $\frac{\Delta t}{t} = \frac{\Delta R}{R}$ ), which yields $3.2\%$ .

Another trap is ignoring the $+1$ in the logarithmic argument, leading to $\frac{\Delta R}{R \ln R}$ .

The correct approach requires differentiating the exact relationship $t = \frac{1}{\lambda} \ln(R+1)$ to find the absolute error $dt$ , and then dividing by $t$ to find the relative error, demonstrating that logarithmic sensitivity heavily suppresses the relative error as $t$ increases.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Activity Units and Transitions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free