To solve this problem, we can use basic trigonometry.

Given:

• The length of the shadow is $25 \, \text{m}$.
• The angle of elevation to the sun is $40^\circ$.

To Find:

• The height of the tree ($h$).

Solution:

We use the tangent function, which is defined as:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

In this problem:

• $\theta = 40^\circ$
• The opposite side is the height of the tree ($h$).
• The adjacent side is the length of the shadow ($25 \, \text{m}$).

So,

$\tan(40^\circ) = \frac{h}{25}$

Solving for $h$:

$h = 25 \times \tan(40^\circ)$

Now, let's calculate:

$h \approx 25 \times 0.8391 \approx 20.98 \, \text{m}$

Answer:

The height of the tree is approximately 21 m.

Would you like further details or help with similar problems?

Here are 5 related questions for further practice:

1. How tall is the tree if the shadow is $30 \, \text{m}$ and the angle of elevation is $45^\circ$?
2. What is the angle of elevation if the tree is $21 \, \text{m}$ tall and the shadow is $25 \, \text{m}$?
3. How long is the shadow if the tree is $25 \, \text{m}$ tall and the angle of elevation is $30^\circ$?
4. Can the height be calculated if only the shadow and the hypotenuse are known?
5. What is the relationship between the length of the shadow and the angle of elevation?

Tip:

For angles less than $45^\circ$, the shadow length will be longer than the height of the object.