Calculating the Length of Wires Supporting a Tower

Imagine a 250-foot tower supported by a single wire running from the ground to the top of the tower at a 70-degree angle of elevation. How do we calculate the length of this wire?

Understanding the Problem

We are given a tower with a height of 250 feet, and a wire is attached from the ground to the very top of the tower, making an angle of 70 degrees with the ground.

Trigonometric Approach

This problem can be solved using basic trigonometric principles. Specifically, we use the sine function, which relates the opposite side of a right triangle to the hypotenuse.

Using the Sine Function

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In this case, the angle of elevation is 70 degrees, and the opposite side (height of the tower) is 250 feet.

Mathematically, we can express this as:

$$ sin(70^{circ}) frac{250 , text{ft}}{L} $$

Solving for ( L ) (the length of the wire), we get:

$$ L frac{250 , text{ft}}{sin(70^{circ})} $$

Calculating the Length

Using a calculator to find the sine of 70 degrees:

$$ sin(70^{circ}) approx 0.9397 $$

Substituting this value into our equation:

$$ L frac{250 , text{ft}}{0.9397} approx 266 , text{ft} $$

Therefore, the length of each wire is approximately 266 feet.

Diagram Visualization

To better understand this, imagine a vertical tower (representing the height of 250 feet) with a wire running from the ground to the top of the tower, forming a 70-degree angle with the ground. This forms a right triangle with the wire as the hypotenuse, the height of the tower as the opposite side, and the distance from the base of the tower to the point where the wire touches the ground as the adjacent side.

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Conclusion

Using trigonometry, particularly the sine function, we calculated that the length of the wire supporting the 250-foot tower at a 70-degree angle of elevation is approximately 266 feet.

Key Takeaways: Use the sine function to relate the opposite side to the hypotenuse in a right triangle. Master basic trigonometric principles to solve real-world problems. Visualize the problem as a right triangle to simplify calculations.