Let's solve this problem step by step, starting by sketching out the situation.

Setup:

• Let the height of the first tower be $h_1$.
• The height of the second tower is given as $h_2 = 60 \, \text{m}$.
• The horizontal distance between the two towers is $140 \, \text{m}$.
• The angle of elevation from the top of the second tower to the top of the first tower is $30^\circ$.

Now, imagine two towers:

• The second tower is shorter and has a height of 60 m.
• We know the angle of elevation from the top of the second tower to the top of the first is $30^\circ$.

Using trigonometry, specifically the tangent function, we can relate the height difference between the two towers and the horizontal distance.

Tangent Formula:

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

In this case:

• The "opposite" is the height difference between the two towers, which is $h_1 - h_2$.
• The "adjacent" is the horizontal distance between them, which is 140 m.
• The angle $\theta = 30^\circ$.

Thus, the equation becomes: $\tan(30^\circ) = \frac{h_1 - 60}{140}$

Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, we substitute it into the equation: $\frac{1}{\sqrt{3}} = \frac{h_1 - 60}{140}$

Now, let's solve for $h_1$.

$h_1 - 60 = \frac{140}{\sqrt{3}}$

$h_1 = 60 + \frac{140}{\sqrt{3}}$

Simplifying this: $h_1 = 60 + \frac{140\sqrt{3}}{3}$

Now let's compute this expression for $h_1$.The height of the first tower is approximately $140.83 \, \text{m}$.

Diagram Explanation:

• The horizontal distance between the two towers is 140 m.
• The second tower has a height of 60 m.
• From the top of the second tower, the angle of elevation to the top of the first tower is $30^\circ$.
• Using trigonometry, we calculated that the height of the first tower is about 140.83 m.

Would you like a diagram, more details, or have any further questions?

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Here are 5 related questions to deepen understanding:

1. How would the result change if the angle of elevation were $45^\circ$?
2. What if the height of the second tower were increased to 80 m?
3. Can you derive the formula for calculating height using trigonometry for any angle?
4. What is the importance of using tangent in right triangle problems?
5. How would the solution change if the distance between towers were doubled?

Tip: Trigonometric functions are especially useful for solving problems involving right triangles and angles of elevation or depression.