This problem can be solved using the concept of similar triangles. The lamp and its shadow form one triangle, and the flagpole and its shadow form another triangle. Since both triangles are formed by the same light source (the sun), they are similar. Therefore, the ratios of the heights to the lengths of the shadows will be equal.

We are given:

• Height of the lamp = 120 cm
• Length of the lamp's shadow = 210 cm
• Length of the flagpole's shadow = 14 m = 1400 cm (converting to cm for consistency)

Let $h_f$ be the height of the flagpole.

Using the proportion for similar triangles: $\frac{\text{Height of the lamp}}{\text{Length of the lamp's shadow}} = \frac{\text{Height of the flagpole}}{\text{Length of the flagpole's shadow}}$ $\frac{120}{210} = \frac{h_f}{1400}$

Now solve for $h_f$: $h_f = \frac{120}{210} \times 1400$

Let's calculate it.The height of the flagpole is 800 cm, or 8 meters.

Would you like further details or have any questions? Here are some related questions for practice:

1. If the lamp's shadow were 280 cm, what would be the height of the flagpole with a shadow of 1400 cm?
2. If the flagpole's shadow is shortened to 10 meters, what would its height be?
3. What is the ratio of the height to the shadow length for both the lamp and the flagpole?
4. If the sun is at a different angle and the lamp’s shadow becomes 315 cm, how tall would the flagpole be with a 14 m shadow?
5. Can this concept of similar triangles be applied if the light source is at a different angle?

Tip: Similar triangles are useful for many problems involving proportional relationships, especially with shadows and heights.