(I) A ball is dropped from a height of 10 m and bounces continuously. With each successive bounce, the ball reaches a height that is $$ 50 \% $$ of the previous height. If this motion continues indefinitely, what is the total vertical distance travelled by the ball over its entire joumey? (g) The diameter of the first circle in an infinite sequence of circles is 12 cm . Each subscquent circle's diameter is two thirds of the diameter of the previous circle. Calculate (1) the total cireumference of all the circles. (2) the total area of the all the circles.

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### **Problem (I): Total Vertical Distance Traveled by the Ball**

**Given:**
- Initial height, $$ h_0 = 10 $$ meters.
- Each bounce reaches $$ 50\% $$ of the previous height.

**Approach:**
1. **Initial Drop:**
   - The ball first drops $$ h_0 = 10 $$ meters.

2. **Subsequent Bounces:**
   - After the first bounce, it rises to $$ h_1 = 0.5 \times h_0 = 5 $$ meters and then drops the same distance.
   - This pattern continues indefinitely, with each bounce reaching half the height of the previous one.

3. **Total Distance Calculation:**
   - **First Drop:** $$ 10 $$ meters.
   - **Bounces:** Each bounce contributes $$ 2 \times h_n $$ meters (up and down).
   - The heights form a geometric series: $$ 5 + 2.5 + 1.25 + \ldots $$
   - **Sum of Infinite Geometric Series:** 
     $$
     \text{Sum} = \frac{a}{1 - r} = \frac{5}{1 - 0.5} = 10 \text{ meters}
     $$
   - **Total Distance:** 
     $$
     10 + 2 \times 10 = 30 \text{ meters}
     $$

**Answer:**
The total vertical distance traveled by the ball is **30 meters**.

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### **Problem (g): Infinite Sequence of Circles**

**Given:**
- Diameter of the first circle, $$ d_1 = 12 $$ cm.
- Each subsequent circle's diameter is $$ \frac{2}{3} $$ of the previous diameter.

**Approach:**

#### **(1) Total Circumference of All Circles**

**Circumference of a Circle:**
$$
C = \pi d
$$

**Sequence of Diameters:**
$$
d_n = 12 \left(\frac{2}{3}\right)^{n-1} \text{ cm}
$$

**Sequence of Circumferences:**
$$
C_n = \pi d_n = 12\pi \left(\frac{2}{3}\right)^{n-1} \text{ cm}
$$

**Sum of Infinite Geometric Series:**
$$
\text{Total Circumference} = \sum_{n=1}^{\infty} C_n = 12\pi \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n = 12\pi \times \frac{1}{1 - \frac{2}{3}} = 36\pi \text{ cm}
$$

#### **(2) Total Area of All Circles**

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

**Sequence of Areas:**
$$
A_n = \frac{\pi d_n^2}{4} = \frac{\pi (12)^2}{4} \left(\frac{2}{3}\right)^{2(n-1)} = 36\pi \left(\frac{4}{9}\right)^{n-1} \text{ cm}^2
$$

**Sum of Infinite Geometric Series:**
$$
\text{Total Area} = \sum_{n=1}^{\infty} A_n = 36\pi \sum_{n=0}^{\infty} \left(\frac{4}{9}\right)^n = 36\pi \times \frac{1}{1 - \frac{4}{9}} = 36\pi \times \frac{9}{5} = \frac{324}{5}\pi \text{ cm}^2
$$

**Answers:**
1. **Total Circumference:** $$ 36\pi $$ centimeters.
2. **Total Area:** $$ \frac{324}{5}\pi $$ square centimeters.

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**Summary of Answers:**

- **Problem (I):** The total vertical distance traveled by the ball is **30 meters**.
- **Problem (g):**
  - **(1) Total Circumference:** $$ 36\pi $$ cm.
  - **(2) Total Area:** $$ \frac{324}{5}\pi $$ cm².
- **Problem (I):** The ball travels a total of **30 meters** vertically. - **Problem (g):** - **(1) Total Circumference:** $$ 36\pi $$ cm. - **(2) Total Area:** $$ \frac{324}{5}\pi $$ cm².

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