The breakdown of a sample of a chemical compounds is represented by the function `p(n)=300(0.5)^(n)`, where p(n) represents the number of millligrams of the substance that remains at the end of n years. Which of the following is true?
I. 300 represents the number of milligrams of the substance that remains after the first year.
II. 0.5 represents the fraction of the starting amount by which the substance gets reduced by the end of each year.
III. Each year the substance gets reduced by one-half of 300.

Options

A. I only
B. II only
C. I and III only
D. II and III only

Answer

To solve the problem, we need to analyze the function \( p(n) = 300(0.5)^n \) and evaluate the three statements provided.

### Step 1: Evaluate Statement I
**Statement I:** 300 represents the number of milligrams of the substance that remains after the first year.

To check this, we need to calculate \( p(1) \):
\[ p(1) = 300(0.5)^1 = 300 \times 0.5 = 150 \text{ mg} \]
Since \( p(1) = 150 \) mg, this means that 300 mg does not represent the amount remaining after the first year.

**Conclusion:** Statement I is false.

### Step 2: Evaluate Statement II
**Statement II:** 0.5 represents the fraction of the starting amount by which the substance gets reduced by the end of each year.

To verify this, we can calculate \( p(2) \):
\[ p(2) = 300(0.5)^2 = 300 \times 0.25 = 75 \text{ mg} \]
Now, we can find the fraction of the substance that remains after the first year and the second year:
- After the first year: \( p(1) = 150 \) mg
- After the second year: \( p(2) = 75 \) mg

Now, we can find the fraction reduced:
\[ \text{Fraction reduced} = 1 - \frac{p(2)}{p(1)} = 1 - \frac{75}{150} = 1 - 0.5 = 0.5 \]
This confirms that 0.5 does represent the fraction by which the substance is reduced each year.

**Conclusion:** Statement II is true.

### Step 3: Evaluate Statement III
**Statement III:** Each year the substance gets reduced by one-half of 300.

To check this, we need to find the difference between the amounts remaining after the first and second years:
\[ \text{Reduction from year 1 to year 2} = p(1) - p(2) = 150 - 75 = 75 \text{ mg} \]
The statement claims that the reduction is half of 300, which is:
\[ \frac{300}{2} = 150 \text{ mg} \]
Since the actual reduction (75 mg) is not equal to 150 mg, this statement is incorrect.

**Conclusion:** Statement III is false.

### Final Conclusion
- Statement I: False
- Statement II: True
- Statement III: False

Thus, the only true statement is **Statement II**.

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