To find the height of the flagpole, we can use the concept of similar triangles. Since the flagpole and Patrick cast shadows under the same conditions, the ratios of their heights to their shadow lengths will be equal.

Given:

• Patrick's height: 1.5 meters
• Patrick's shadow: 1 meter
• Flagpole's shadow: 4.5 meters

Let $h$ be the height of the flagpole. The ratio can be set up as follows:

$\frac{\text{Patrick's height}}{\text{Patrick's shadow}} = \frac{\text{Flagpole's height}}{\text{Flagpole's shadow}}$

$\frac{1.5}{1} = \frac{h}{4.5}$

Now, solve for $h$:

$h = 1.5 \times 4.5 = 6.75 \text{ meters}$

Therefore, the height of the flagpole is 6.75 meters.

Would you like more details, or do you have any further questions?

Here are some related questions:

1. How can we use ratios and proportions to solve real-life problems?
2. What are some other examples of using similar triangles in practical situations?
3. How does the concept of shadow length change with different light sources?
4. Can you solve for the height of a flagpole if Patrick was standing on a slope?
5. What is the importance of understanding similar triangles in geometry?

Tip: When dealing with similar triangles, always check if they share the same angles to ensure the ratio approach is valid.