This problem can be solved using the Pythagorean Theorem. The rope forms the hypotenuse of a right triangle, with the height of the flagpole as one leg and the distance from the base of the flagpole to the hook as the other leg.

The Pythagorean Theorem states: $a^2 + b^2 = c^2$ Where:

• $a$ is the height of the flagpole,
• $b = 4.3 \, \text{m}$ is the distance from the base of the flagpole to the hook,
• $c = 7.8 \, \text{m}$ is the length of the rope (the hypotenuse).

We need to solve for $a$: $a^2 + 4.3^2 = 7.8^2$ $a^2 + 18.49 = 60.84$ $a^2 = 60.84 - 18.49 = 42.35$ $a = \sqrt{42.35} \approx 6.5 \, \text{m}$

So, the height of the flagpole is approximately 6.5 meters.

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How would the height change if the rope length increased to 10 meters?
2. How do you determine which side of a triangle is the hypotenuse?
3. Can the rope be shorter than the distance from the hook to the flagpole base?
4. How is the Pythagorean theorem applied to non-right triangles?
5. If the distance between the hook and the flagpole base is doubled, how does the height change?

Tip: In word problems, identifying right triangles helps apply the Pythagorean Theorem correctly.