The number of ²³⁸ U atoms in an ancient rock equals the number of ²⁰⁶ Pb atoms. The half-life of decay of ²³⁸ U is 4.5x10⁹ years. Estimate the age of the rock assuming that all the ²⁰⁶ Pb atoms are formed from the decay of ²³⁸U.

Explanation

To estimate the age of the rock based on the given information, we can follow these steps:

### Step 1: Understand the relationship between Uranium and Lead
We know that the number of \(^{238}U\) atoms equals the number of \(^{206}Pb\) atoms in the rock. This means that all the lead atoms present in the rock have been formed from the decay of uranium atoms.

### Step 2: Set up the initial conditions
Let \(N_0\) be the initial number of \(^{238}U\) atoms at time \(t = 0\). After some time \(t\), the number of \(^{238}U\) atoms remaining will be \(N\), and the number of \(^{206}Pb\) atoms formed will be \(N_0 - N\).

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