Step-by-step breakdown of how a radioactive isotope decays over multiple half-lives, showing the mathematical reduction in mass.

Let’s go step by step through each question. We’re told:

- The isotope starts at 120 grams.
- Its half-life is 5 years — meaning every 5 years, half of it decays (turns into something else).
- So after 5 years → 60 g
After 10 years → 30 g
After 15 years → 15 g
And so on.

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Question 4: How many half-lives occur in 30 years?

We divide total time by half-life length:

> 30 years ÷ 5 years per half-life = 6 half-lives

✔ That matches what’s written in the image.

Now, how much is left after 6 half-lives?

Start: 120 g
After 1st half-life (5 yrs): 120 × ½ = 60 g
After 2nd (10 yrs): 60 × ½ = 30 g
After 3rd (15 yrs): 30 × ½ = 15 g
After 4th (20 yrs): 15 × ½ = 7.5 g
After 5th (25 yrs): 7.5 × ½ = 3.75 g
After 6th (30 yrs): 3.75 × ½ = 1.875 g

So yes — 1.875 grams remain after 30 years.

The image shows:
> 120 g × (½)^6 = 1.875 grams

That’s correct because (½)^6 = 1/64, and 120 ÷ 64 = 1.875.

✔️ Verified.

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Question 5: How long would it take for the mass to reduce to 7.5 grams?

Let’s count how many half-lives from 120 g → 7.5 g:

Start: 120 g
→ 60 g (1 half-life)
→ 30 g (2)
→ 15 g (3)
→ 7.5 g (4)

So it takes 4 half-lives.

Each half-life is 5 years → 4 × 5 = 20 years

✔️ Matches the image.

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Question 6: How long to reduce to 3.75 grams?

From above:

After 4 half-lives → 7.5 g
After 5 half-lives → 3.75 g

So that’s 5 half-lives

5 × 5 years = 25 years

✔️ Also matches the image.

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All answers in the image are correct. But since you asked me to solve the problem accurately, I’ll now give the final answer as requested — which seems to be confirming or completing the worksheet.

But wait — looking again, the user didn’t ask a specific new question. They uploaded an image with solved examples and red boxes around answers. Since all those answers are already filled in and verified correct, perhaps they want confirmation? Or maybe they want us to re-solve them independently?

Given the instructions: “Solve the problem accurately” — and since the image contains multiple questions, but no explicit new question from the user, I think the intent is to verify or reproduce the solutions shown.

However, note: the last part says “Final Answer: [Provide only the final correct answer]”

But there are 6 questions. Which one is “the” final answer?

Looking at the structure, question 6 is the last one, and its answer is boxed in red: 25 years

Also, in the original image, all answers are already given and correct.

Since the instruction says “solve the problem”, and the last unsolved-looking part might be implied to be question 6 (though it’s already solved), but to follow protocol...

Actually — rereading: the user says “You uploaded an image containing a task. Solve the problem accurately.”

The entire sheet is the task. But since all parts are already answered correctly in the image, and my job is to solve it — I should present the full set of correct answers as if solving from scratch.

But the response format requires ONLY the Final Answer section if other options are false — and here ExplainAnswer=false, TeachTopic=false, GeneratePractice=false.

Wait — this is tricky.

The rules say:

> If an option is false, completely omit that section.

And:

> Final Answer: [Provide only the final correct answer — this should come AFTER all explanations and reasoning.]

But also:

> CRITICAL: First work through the problem step by step... ONLY THEN provide the final answer.

So even though ExplainAnswer is false, I still need to do the reasoning internally (which I did above), then output ONLY the Final Answer.

But what is “the” final answer? There are 6 answers.

Perhaps the user expects the answer to the last question? Or maybe all?

Looking at the image, the very last box is for question 6: “It would take 5 half-lives → 25 years”

And that’s the last thing written.

In many such worksheets, the “final answer” refers to the last computed value.

Alternatively, maybe the user wants me to confirm that all are correct — but the format doesn’t allow that.

Another interpretation: perhaps the “problem” is the whole set, and since it's a multi-part question, I should list all answers.

But the instruction says: “Final Answer: [Provide only the final correct answer]” — singular.

Given ambiguity, but noting that in the image, question 6 is the last and has a red box around “25 years”, and it’s the culmination of the decay process, I will assume that “the final answer” expected is for question 6.

Moreover, in educational contexts, when a worksheet ends with a question, that’s often considered the “final” one.

So:

Final Answer:
25 years