We can solve this problem using the Pythagorean theorem, which states:

$a^2 + b^2 = c^2$

where:

• $a$ is the height of the flagpole,
• $b$ is the length of the shadow (32 meters),
• $c$ is the hypotenuse (40 meters).

We need to find the height of the flagpole ($a$).

Plugging the values into the equation:

$a^2 + 32^2 = 40^2$

$a^2 + 1024 = 1600$

Subtract 1024 from both sides:

$a^2 = 1600 - 1024$

$a^2 = 576$

Now, take the square root of both sides:

$a = \sqrt{576} = 24$

Thus, the height of the flagpole is 24 meters.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How would the height of the flagpole change if the shadow length increased?
2. Can you apply the Pythagorean theorem to find the height if the shadow is unknown but the height and hypotenuse are given?
3. What would happen to the height if the hypotenuse were shorter?
4. How do you calculate the angle of elevation of the sun in this scenario?
5. Can trigonometry be used to find the height without using the Pythagorean theorem?

Tip: Always label the sides of a triangle carefully when applying the Pythagorean theorem!