A vertical pole 5 m 60 cm high casts a shadow 3 m 20 cm long. Find at the same time (i) the length of shadown cast by another pole 10 m 50 cm high.
(ii) the height of a pole which casts a shadow 5 m long.

Answer

To solve the problem step by step, we will use the concept of direct proportion, where the height of the pole and the length of the shadow are directly proportional to each other.

### Given:
1. Height of the first pole = 5 m 60 cm = 560 cm
2. Length of the shadow of the first pole = 3 m 20 cm = 320 cm
3. Height of the second pole = 10 m 50 cm = 1050 cm
4. Length of the shadow of the second pole = L (to be found)
5. Length of the shadow of the third pole = 5 m = 500 cm
6. Height of the third pole = H (to be found)

### Part (i): Finding the length of the shadow cast by another pole 10 m 50 cm high.

1. **Set up the proportion:**
\[ \frac{\text{Height of first pole}}{\text{Length of shadow of first pole}} = \frac{\text{Height of second pole}}{\text{Length of shadow of second pole}} \]
\[ \frac{560}{320} = \frac{1050}{L} \]

2. **Cross-multiply to solve for L:**
\[ 560 \cdot L = 320 \cdot 1050 \]

3. **Calculate \(320 \cdot 1050\):**
\[ 320 \cdot 1050 = 336000 \]

4. **Rearranging the equation:**
\[ L = \frac{336000}{560} \]

5. **Calculate \(L\):**
\[ L = 600 \text{ cm} \]

### Part (ii): Finding the height of a pole which casts a shadow 5 m long.

1. **Set up the proportion:**
\[ \frac{\text{Height of first pole}}{\text{Length of shadow of first pole}} = \frac{\text{Height of third pole}}{\text{Length of shadow of third pole}} \]
\[ \frac{560}{320} = \frac{H}{500} \]

2. **Cross-multiply to solve for H:**
\[ 560 \cdot 500 = 320 \cdot H \]

3. **Calculate \(560 \cdot 500\):**
\[ 560 \cdot 500 = 280000 \]

4. **Rearranging the equation:**
\[ H = \frac{280000}{320} \]

5. **Calculate \(H\):**
\[ H = 875 \text{ cm} \]

### Final Answers:
- (i) The length of the shadow cast by the pole 10 m 50 cm high is **600 cm**.
- (ii) The height of the pole which casts a shadow 5 m long is **875 cm**.