
The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. Starting with 115 grams of a radioactive isotope, how much will be left after 3 half-lives?
round your answer to the nearest gram.

The concept of half-life tells us that with each half-life, the amount of a radioactive isotope decreases to half of its previous amount.

Given:
- Initial mass of the isotope (\(M_0\)) = 115 grams
- Number of half-lives (\(n\)) = 3

To find the remaining mass (\(M\)) after 3 half-lives, we use the formula:

\[
M = M_0 \times \left(\frac{1}{2}\right)^n
\]

Substitute the values:

\[
M = 115 \times \left(\frac{1}{2}\right)^3
\]

Calculating step-by-step:

1. Calculate \(\left(\frac{1}{2}\right)^3\):

\[
\left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]

2. Multiply by the initial mass:

\[
M = 115 \times \frac{1}{8} = \frac{115}{8}
\]

3. Simplify the division:

\[
\frac{115}{8} = 14.375
\]

Rounded to the nearest gram:

\[
M \approx 14 \text{ grams}
\]

So, after 3 half-lives, approximately **14 grams** of the radioactive isotope will be left.

---

Would you like further details or have any questions?

Here are some related questions to consider:
1. What is the remaining mass after 5 half-lives?
2. How does the concept of half-life apply to carbon dating?
3. If the half-life of a substance is 2 years, how long will it take for the substance to reduce to 1/16th of its initial mass?
4. What factors can affect the half-life of a radioactive isotope?
5. How would you graph the decay of a radioactive isotope over time?

**Tip:** Understanding the exponential decay function is crucial for grasping how quantities diminish over time in fields like chemistry and physics.

Learn how to calculate the remaining mass of a radioactive isotope after three half-lives. Starting with 115 grams, understand the concept of exponential decay and apply it to solve this problem. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation