To solve this problem, we need to find the shortest length of wire needed to connect the top of a 140-ft tower to a point that is 55 feet downhill from the base of the tower. The mountain side has an inclination of $32^\circ$ relative to the horizontal.

We can solve this by applying trigonometry in a right triangle. Here’s a step-by-step approach:

Step 1: Identify the Components in the Triangle

Let:

• $h = 140$ ft be the height of the tower.
• $d = 55$ ft be the horizontal distance along the incline from the base of the tower.
• The angle of incline is $32^\circ$.

We are looking for the hypotenuse (the shortest length of the wire), which we'll denote as $L$.

Step 2: Set Up the Right Triangle

In this scenario:

1. The height of the triangle formed is $h = 140$ ft (from the base to the top of the tower).
2. The base of the triangle is the horizontal distance $d = 55$ ft, adjusted by the incline angle of $32^\circ$.

Step 3: Use the Law of Cosines

To find the length $L$ of the wire, we can use the Law of Cosines because we have two sides (140 ft and 55 ft) and the angle between them ($32^\circ$):

$L^2 = h^2 + d^2 - 2 \cdot h \cdot d \cdot \cos(32^\circ)$

Plugging in the values: $L^2 = 140^2 + 55^2 - 2 \cdot 140 \cdot 55 \cdot \cos(32^\circ)$

Calculating each term:

1. $140^2 = 19600$
2. $55^2 = 3025$
3. $2 \cdot 140 \cdot 55 \cdot \cos(32^\circ)$

Using $\cos(32^\circ) \approx 0.848$: $L^2 = 19600 + 3025 - 2 \cdot 140 \cdot 55 \cdot 0.848$ $L^2 = 19600 + 3025 - 13024$ $L^2 = 9581$

Taking the square root to find $L$: $L \approx \sqrt{9581} \approx 98 \text{ ft}$

Answer

The shortest length of wire needed is approximately 98 feet (rounded to the nearest foot).

Would you like further clarification on any step, or need a detailed breakdown?

Here are five related questions to further your understanding:

1. How would the solution change if the mountain incline angle were different?
2. What if the distance from the base of the tower were increased?
3. How can we find the angle between the tower and the ground using trigonometry?
4. How would you solve for the wire length if given only the incline and height?
5. How does the Law of Cosines help in finding the length in non-right triangles?

Tip: The Law of Cosines is useful for finding unknown sides or angles when dealing with oblique (non-right) triangles.