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The image contains two exercises, numbered 46 and 47, related to the historical approximations of π (pi).

### Exercise 46:
It references Archimedes' approximation of \(\pi\), around 250 B.C., with the following inequality:
\[
\frac{223}{71} < \pi < \frac{22}{7}
\]
The questions are:
1. **a)** Is this a decimal approximation?
2. **b)** Determine the amplitude of this approximation. Round to the nearest thousandth.

### Exercise 47:
This exercise discusses historical approximations of π in India and China.
1. **a)** In India, around 380 B.C., the approximation \( 3 + \frac{177}{1250} \) was used for \(\pi\). The instruction asks to copy and complete:
   - "This value is the rounded approximation of π to 10^(-...)"
   
2. **b)** In China, in the 5th century, the approximation \( \frac{355}{113} \) was used for \(\pi\). The instruction again asks to copy and complete:
   - "This value allows to know the rounded approximation of π to 10^(-...)"

---

### Solutions:

#### Exercise 46:
- **a)** To check if this is a decimal approximation, we convert the fractions to decimal values:
  \[
  \frac{223}{71} \approx 3.140845 \quad \text{and} \quad \frac{22}{7} \approx 3.142857
  \]
  So the range is \( 3.140845 < \pi < 3.142857 \), which is indeed a decimal approximation.

- **b)** To find the amplitude of this range, subtract the lower bound from the upper bound:
  \[
  3.142857 - 3.140845 = 0.002012
  \]
  The amplitude is approximately **0.002** (rounded to the nearest thousandth).

#### Exercise 47:
- **a)** First, convert the Indian approximation \( 3 + \frac{177}{1250} \) to decimal:
  \[
  3 + \frac{177}{1250} = 3 + 0.1416 = 3.1416
  \]
  This is π rounded to \(10^{-4}\) (four decimal places).

- **b)** For the Chinese approximation \( \frac{355}{113} \):
  \[
  \frac{355}{113} \approx 3.14159292
  \]
  This value approximates π to \(10^{-6}\) (six decimal places).

---

Let me know if you need further details or explanations! Here are 5 related questions to expand on this topic:

1. What is the exact value of \(\pi\) to six decimal places?
2. Can you explain how Archimedes derived his approximation of \(\pi\)?
3. What other ancient cultures contributed to the approximation of \(\pi\)?
4. How does the Chinese approximation compare with the modern value of \(\pi\)?
5. What methods are used today to calculate \(\pi\) to millions of decimal places?

**Tip:** Understanding how \(\pi\) was approximated historically gives insight into early mathematical methods.

Exercise 46: Archimedes' approximation of π with fractions 223/71 and 22/7. Questions include checking if it's a decimal approximation and finding the amplitude.
Exercise 47: Historical approximations of π in India and China with fractions 3 + 177/1250 and 355/113 respectively. Questions ask to determine the rounded approximation to a certain decimal place.

Explore historical approximations of π, including Archimedes' bounds and values used in ancient India and China. Learn how to verify if an approximation is decimal and calculate amplitude, suitable for students in Grades 9-10.

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