At a certain time a tree 6 m high casts a shadow of length 8 meters. At the same time a pole casts a shadow of length 20 meters. Find the height of the pole.

Answer

To solve the problem step by step, we will use the concept of ratios and proportions.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Height of the tree = 6 meters
- Length of the shadow of the tree = 8 meters
- Length of the shadow of the pole = 20 meters
- Height of the pole = X meters (we need to find this)

2. **Set Up the Proportion:**
- The ratio of the height of the tree to the length of its shadow is equal to the ratio of the height of the pole to the length of its shadow.
- This can be represented as:
\[ \frac{\text{Height of Tree}}{\text{Length of Tree's Shadow}} = \frac{\text{Height of Pole}}{\text{Length of Pole's Shadow}} \]
- Substituting the known values:
\[ \frac{6}{8} = \frac{X}{20} \]

3. **Cross Multiply:**
- To eliminate the fractions, we cross-multiply:
\[ 6 \times 20 = 8 \times X \]
- This simplifies to:
\[ 120 = 8X \]

4. **Solve for X:**
- To find the height of the pole (X), divide both sides of the equation by 8:
\[ X = \frac{120}{8} \]
- Performing the division:
\[ X = 15 \text{ meters} \]

5. **Conclusion:**
- The height of the pole is 15 meters.