QuestionJune 26, 2025

A radioactive isotope has a half-life of 1 billion years. A rock originally contained 50 grams of radioactive isotope, and now contains 6.25 grams. Approximately how many years old is the rock? 125,000 years 2 billion years 3 billon years 125 million years 8 billion years

A radioactive isotope has a half-life of 1 billion years. A rock originally contained 50 grams of radioactive isotope, and now contains 6.25 grams. Approximately how many years old is the rock? 125,000 years 2 billion years 3 billon years 125 million years 8 billion years
A radioactive isotope has a half-life of 1 billion years. A rock originally contained 50 grams of radioactive isotope, and now contains 6.25 grams. Approximately how many years old is the rock? 125,000 years 2 billion years 3 billon years 125 million years 8 billion years

Solution
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Answer

3 billion years Explanation 1. Determine the number of half-lives Use the formula for half-life decay: N = N_0 \times (0.5)^{t/T}, where N is the remaining amount, N_0 is the initial amount, t is time, and T is the half-life. Solve for t: 6.25 = 50 \times (0.5)^{t/1}. Simplify to find (0.5)^{t} = \frac{6.25}{50} = 0.125. 2. Calculate the number of half-lives Recognize that 0.125 = (0.5)^3. Therefore, t = 3 half-lives. 3. Calculate the age of the rock Multiply the number of half-lives by the half-life duration: 3 \times 1 billion years = 3 billion years.

Explanation

1. Determine the number of half-lives<br /> Use the formula for half-life decay: $N = N_0 \times (0.5)^{t/T}$, where $N$ is the remaining amount, $N_0$ is the initial amount, $t$ is time, and $T$ is the half-life. Solve for $t$: $6.25 = 50 \times (0.5)^{t/1}$. Simplify to find $(0.5)^{t} = \frac{6.25}{50} = 0.125$.<br /><br />2. Calculate the number of half-lives<br /> Recognize that $0.125 = (0.5)^3$. Therefore, $t = 3$ half-lives.<br /><br />3. Calculate the age of the rock<br /> Multiply the number of half-lives by the half-life duration: $3 \times 1$ billion years = 3 billion years.
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