Question 26: To determine how many half-lives it takes for a radioactive substance to decay to one-eighth of its original amount, we can use the formula:
- After \( n \) half-lives, the remaining amount is given by: \[ \text{Remaining amount} = \frac{\text{Original amount}}{2^n} \]
We want to find \( n \) such that:
\[ \frac{\text{Original amount}}{2^n} = \frac{1}{8} \times \text{Original amount} \]
This simplifies to:
\[ \frac{1}{2^n} = \frac{1}{8} \]
Since \( \frac{1}{8} = \frac{1}{2^3} \), we have:
\[ 2^n = 2^3 \]
This gives us \( n = 3 \).
So, the answer to Question 26 is a. three.
Question 27: The half-life of the radioactive element X is 5 days. We want to find out how much remains after 10 days.
After each half-life:
- After 5 days (1 half-life): \( 100 , \text{g} \) → \( 50 , \text{g} \)
- After another 5 days (2 half-lives): \( 50 , \text{g} \) → \( 25 , \text{g} \)
So, after 10 days, the amount of element X remaining is 25 grams.
Thus, the answer to Question 27 is c. 25 g.
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