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

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Let's analyze the problem step by step.

### Given:
1. **First Circle:**
   - **Diameter ($$d_1$$)**: 12 cm
   - **Radius ($$r_1$$)**: $$ \frac{d_1}{2} = 6 $$ cm

2. **Subsequent Circles:**
   - Each circle's **diameter** is $$\frac{2}{3}$$ of the previous circle's diameter.
   - This creates a **geometric sequence** for diameters with a common ratio ($$r$$) of $$\frac{2}{3}$$.

### 1. Total Circumference of All Circles

**Circumference of a Single Circle:**
$$ C_n = \pi \cdot d_n $$

**Given the sequence of diameters:**
$$ d_n = 12 \left(\frac{2}{3}\right)^{n-1} $$

**Total Circumference ($$C_{\text{total}}$$):**
$$ 
C_{\text{total}} = \sum_{n=1}^{\infty} C_n = \sum_{n=1}^{\infty} \pi \cdot 12 \left(\frac{2}{3}\right)^{n-1} 
$$
Factor out $$\pi \cdot 12$$:
$$ 
C_{\text{total}} = 12\pi \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n 
$$
This is a geometric series with sum:
$$ 
\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n = \frac{1}{1 - \frac{2}{3}} = 3 
$$
Thus:
$$ 
C_{\text{total}} = 12\pi \cdot 3 = 36\pi \text{ cm}
$$

### 2. Total Area of All Circles

**Area of a Single Circle:**
$$ A_n = \pi r_n^2 = \pi \left(\frac{d_n}{2}\right)^2 = \frac{\pi d_n^2}{4} $$

**Given the sequence of diameters:**
$$ d_n = 12 \left(\frac{2}{3}\right)^{n-1} $$

**Total Area ($$A_{\text{total}}$$):**
$$ 
A_{\text{total}} = \sum_{n=1}^{\infty} A_n = \sum_{n=1}^{\infty} \frac{\pi}{4} \cdot \left[12 \left(\frac{2}{3}\right)^{n-1}\right]^2 
$$
Simplify inside the sum:
$$ 
A_{\text{total}} = \frac{\pi}{4} \cdot 144 \sum_{n=0}^{\infty} \left(\frac{4}{9}\right)^n 
$$
This is a geometric series with sum:
$$ 
\sum_{n=0}^{\infty} \left(\frac{4}{9}\right)^n = \frac{1}{1 - \frac{4}{9}} = \frac{9}{5} 
$$
Thus:
$$ 
A_{\text{total}} = \frac{\pi}{4} \cdot 144 \cdot \frac{9}{5} = \frac{324}{5}\pi \text{ cm}^2 
$$

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

**Numerically:**
- $$ 36\pi \approx 113.10 $$ cm
- $$ \frac{324}{5}\pi \approx 203.20 $$ cm²

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

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

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