Certainly! Here’s a **step-by-step explanation** for how to solve the problem in the image: --- ### **Step 1: Understand the Problem** - You have a zircon crystal containing Uranium-235 (U-235) and Lead-207 (Pb-207). - You are given: - **6,000 atoms of U-235** (parent isotope) - **18,000 atoms of Pb-207** (daughter isotope) - The **half-life** of U-235 decaying to Pb-207 is **approximately 700 million years**. - You need to **calculate the age of the zircon**. --- ### **Step 2: Set Up the Decay Formula** The age dating formula is: \[ \text{Number of half-lives passed} \times \text{Half-life period} = \text{Age of the mineral} \] The **fraction of parent atoms remaining** is: \[ \frac{P}{P + D} \times 100 \] Where: - \( P \) = Number of parent atoms remaining (\(U\text{-}235 = 6,000\)) - \( D \) = Number of daughter atoms present (\(Pb\text{-}207 = 18,000\)) --- ### **Step 3: Calculate the Number of Half-lives Passed** First, calculate the original number of U-235 atoms: \[ \text{Original U-235 atoms} = P + D = 6,000 + 18,000 = 24,000 \] Fraction of U-235 remaining: \[ \text{Fraction remaining} = \frac{6,000}{24,000} = 0.25 \] This means **25% of the U-235 remains**. --- ### **Step 4: Determine the Number of Half-lives** Use the decay equation: \[ \left(\frac{1}{2}\right)^{n} = \text{Fraction remaining} \] Where \( n \) = number of half-lives. \[ \left(\frac{1}{2}\right)^{n} = 0.25 \] \[ 0.25 = \left(\frac{1}{2}\right)^2 \] So, \[ n = 2 \] --- ### **Step 5: Calculate the Age** \[ \text{Age} = n \times \text{half-life} \] \[ \text{Age} = 2 \times 700\, \text{million years} = 1,400\, \text{million years} \] --- ### **Step 6: Answer the Conceptual Question** **Does the zircon age represent the timing of metamorphism or the age of formation of the original rock?** **Brief Explanation:** The zircon age represents the time **when the zircon crystal itself formed** (usually during crystallization from melt in igneous rocks or during high-grade metamorphism if new zircon grows). If the zircon survived metamorphism without losing its U-Pb system, the age will reflect the time of original formation, **not the metamorphism**. If new zircon grew during metamorphism, the age would date the metamorphic event. --- ## **Summary Table** | Step | Calculation/Concept | Result | |------|---------------------|-----------------------| | 1 | Total (P + D) | 24,000 | | 2 | Fraction remaining | 0.25 | | 3 | Half-lives (n) | 2 | | 4 | Age | 1,400 million years | --- ## **Final Answers** ### **1. Age of zircon:** **1,400 million years** ### **2. Does the age represent timing of metamorphism?** *No, unless new zircon formed during metamorphism. Usually, it represents the original crystallization age of the zircon unless the system was reset during metamorphism.* --- If you need this written out or formatted differently, let me know!