**Understanding Half-Life Calculation using Synthetic Isotopes**Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 grams.**Question:** What is the half-life of this isotope?**Multiple Choice Options:**- 1.5 minutes- 2.25 minutes- 8.5 minutes- 4.25 minutes**Explanation:**To solve for the half-life of an isotope, we use the concept of exponential decay. The half-life is the time it takes for a substance to reduce to half its initial amount. For this isotope, we need to determine the number of half-lives it has undergone to decay from 128 grams to 2 grams in 25.5 minutes.1. Start with the initial mass of the isotope: 128 grams.2. After one half-life, the mass would be: \( \frac{128}{2} = 64 \) grams.3. Continuing this process, we get: - Second half-life: \( \frac{64}{2} = 32 \) grams - Third half-life: \( \frac{32}{2} = 16 \) grams - Fourth half-life: \( \frac{16}{2} = 8 \) grams - Fifth half-life: \( \frac{8}{2} = 4 \) grams - Sixth half-life: \( \frac{4}{2} = 2 \) gramsSince it took 25.5 minutes to decay through 6 half-lives, we divide the total time by the number of half-lives to find the duration of one half-life:\[ \text{Half-life} = \frac{25.5 \text{ minutes}}{6} = 4.25 \text{ minutes} \]Thus, based on the calculation, the half-life of this synthetic isotope is 4.25 minutes. The correct answer is the last option:- 4.25 minutes

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Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 g. What is the half-life of this isotope? O 1.5 minutes O 2.25 minutes O 8.5 minutes O 4.25 minutes

Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 g. What is the half-life of this isotope? O 1.5 minutes O 2.25 minutes O 8.5 minutes O 4.25 minutes

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter5: Exponential And Logarithmic Functions

Section5.2: Applications Of Exponential Functions

Problem 3E: Use a calculator to help solve each problem. Tritium decay Tritium, a radioactive isotope of...

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