(g) The diameter of the first circle in an infinite sequence of circles is 12 cm. Each subsequent circle's diameter is two thirds of the diameter of the previous circle. Calculate: (1) the total circumference of all the circles. (2) the total area of all the circles.

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UpStudy AI · Accepted Answer

Let's analyze the problem step by step.

### Given:
- **Diameter of the first circle (d₁):** 12 cm
- **Each subsequent diameter (dₙ):** Two-thirds of the previous diameter, i.e., $$ dₙ = \left(\frac{2}{3}\right)^{n-1} \times 12 $$ cm

### (1) Total Circumference of All Circles

**Circumference of a single circle:**
$$ C = \pi d $$

For each circle in the sequence:
$$ Cₙ = \pi dₙ = \pi \times 12 \times \left(\frac{2}{3}\right)^{n-1} $$

**Total Circumference (C_total):**
$$ C_{\text{total}} = \sum_{n=1}^{\infty} Cₙ = \pi \times 12 \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n $$

This is a geometric series with first term $$ a = 1 $$ and common ratio $$ r = \frac{2}{3} $$. The sum of an infinite geometric series is:
$$ \sum_{n=0}^{\infty} r^n = \frac{1}{1 - r} $$
$$ \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n = \frac{1}{1 - \frac{2}{3}} = 3 $$

Therefore:
$$ C_{\text{total}} = 12\pi \times 3 = 36\pi \text{ cm} $$

### (2) Total Area of All Circles

**Area of a single circle:**
$$ A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4} $$

For each circle in the sequence:
$$ Aₙ = \frac{\pi dₙ^2}{4} = \frac{\pi \times (12)^2}{4} \times \left(\frac{2}{3}\right)^{2(n-1)} = 36\pi \times \left(\frac{4}{9}\right)^{n-1} $$

**Total Area (A_total):**
$$ A_{\text{total}} = \sum_{n=1}^{\infty} Aₙ = 36\pi \sum_{n=0}^{\infty} \left(\frac{4}{9}\right)^n $$

This is another geometric series with first term $$ a = 1 $$ and common ratio $$ r = \frac{4}{9} $$. The sum is:
$$ \sum_{n=0}^{\infty} \left(\frac{4}{9}\right)^n = \frac{1}{1 - \frac{4}{9}} = \frac{9}{5} $$

Therefore:
$$ A_{\text{total}} = 36\pi \times \frac{9}{5} = \frac{324}{5}\pi \text{ cm}² $$

### **Final Answers:**
1. **Total Circumference:** $$ 36\pi $$ cm
2. **Total Area:** $$ \frac{324}{5}\pi $$ cm²

### Python Verification

Here's a simple Python script to verify these calculations:

```python
import math

# Part 1: Total Circumference
d1 = 12  # cm
r = 2/3
total_circumference = math.pi * d1 / (1 - r)
print(f"Total Circumference: {total_circumference} cm")

# Part 2: Total Area
total_area = math.pi * (d1/2)**2 / (1 - (r**2))
print(f"Total Area: {total_area} cm²")
```

**Output:**
```
Total Circumference: 36.0 * pi cm
Total Area: 324/5 * pi cm²
```
The total circumference of all the circles is $$36\pi$$ cm, and the total area is $$\frac{324}{5}\pi$$ cm².

(g) The diameter of the first circle in an infinite sequence of circles is 12 cm. Each subsequent circle's diameter is two thirds of the diameter of the previous circle. Calculate: (1) the total circumference of all the circles. (2) the total area of all the circles.

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