Question

An isotope has a half-life of 42 years. What fraction of the original quantity remains after 294 years?

A. 1/64
B. 1/128
C. 1/256
D. 1/512

Answer 1

The fraction of the original quantity that remains after 294 years for an isotope with a half-life of 42 years is 1/128.

Step-by-step explanation:

The fraction of the original quantity that remains after 294 years can be determined by dividing the total time elapsed by the half-life of the isotope. In this case, 294 years divided by 42 years (the half-life) gives us 7. The exponent n in the equation (1/2)^n represents the number of half-lives, and since we have 7 half-lives, the fraction that remains is (1/2)^7, which is equal to 1/128. Therefore, the answer is Option B: 1/128.

Answer 2

After 294 years, which represents 7 half-lives for an isotope with a half-life of 42 years, the fraction of the original quantity that remains is 1/128.

This correct answer is B.

Step-by-step explanation:

The question asks us to calculate what fraction of a radioactive isotope remains after a certain number of half-lives have passed. In this instance, an isotope has a half-life of 42 years, and we are to determine the remaining fraction after 294 years.

To do this calculation, we need to find out how many half-lives 294 years represents. We divide 294 by the half-life period:

294 years ÷ 42 years/half-life = 7 half-lives

After each half-life, half of the remaining isotope decays. This can be represented as:

• 1st half-life: 1/2 remains
• 2nd half-life: (1/2)² remains
• 3rd half-life: (1/2)³ remains
• ...
• 7th half-life: (1/2)⁷ remains

We calculate the remaining fraction after 7 half-lives:

(1/2)⁷ = 1/128

Therefore, the fraction of the isotope that remains after 294 years is 1/128, corresponding to answer choice B.

This correct answer is B.