Let's break the problem into steps to find the height of the second tree using trigonometry.

Step 1: Assign variables

• Let the height of the first tree be $h_1 = 13 \, \text{m}$.
• Let the height of the second tree be $h_2$.
• The angle of elevation to the top of the second tree is $45^\circ$.
• The angle of depression to the bottom of the second tree is $30^\circ$.

We will assume both trees are separated by a horizontal distance $d$, which we need to find.

Step 2: Use the angle of depression (30°)

From the top of the first tree, the angle of depression to the bottom of the second tree is $30^\circ$. The vertical distance from the top of the first tree to the bottom of the second tree is $h_1$.

By the tangent function in trigonometry: $\tan(30^\circ) = \frac{h_1}{d}$ $\tan(30^\circ) = \frac{13}{d}$ Using $\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{1}{1.732}$: $\frac{1}{1.732} = \frac{13}{d}$ $d = 13 \times 1.732 = 22.516 \, \text{m}$

Step 3: Use the angle of elevation (45°)

Now, consider the angle of elevation from the top of the first tree to the top of the second tree. This gives the following relationship: $\tan(45^\circ) = \frac{h_2 - h_1}{d}$ Since $\tan(45^\circ) = 1$, we have: $1 = \frac{h_2 - 13}{22.516}$ $h_2 - 13 = 22.516$ $h_2 = 22.516 + 13 = 35.516 \, \text{m}$

Conclusion

The height of the second tree is approximately 35.52 meters.

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Do you have any questions, or would you like further clarification?

Related questions:

1. How can we calculate angles of elevation or depression in practical situations?
2. What is the significance of the tangent function in trigonometric problems?
3. How do different trigonometric ratios (sine, cosine, tangent) apply to right-angled triangles?
4. What methods are used to measure the height of tall objects indirectly?
5. How would the problem change if the angle of elevation were 60° instead of 45°?

Tip:

Always draw a diagram when dealing with trigonometric problems involving angles of elevation or depression—it makes visualization and problem-solving easier!