Problem 50 - Algebra and Trigonometry

Question

A 125-ft tower is located on the side of a mountain that is inclined \(32^{\circ}\) to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 \(\mathrm{ft}\) downhill from the base of the tower. Find the shortest length of wire needed.

Step-by-Step Solution

Answer

The shortest length of wire needed is approximately 156 feet.

Step 1: Visualize the Problem and Identify Given Information

Picture the problem as a right triangle. The tower is vertical, and the downhill section forms an inclined line at a 32° angle to the horizontal. The guy wire forms the hypotenuse of this triangle. The height of the tower is 125 ft, and the base downhill distance is 55 ft.

Step 2: Adjust for Inclination

Calculate the effective horizontal distance. Since the ground is inclined, we need to consider the horizontal component of this distance. The effective horizontal distance can be found using the cosine of the inclination angle: \( \text{horizontal distance} = 55 \times \cos(32^{\circ}) \).

Step 3: Calculate Total Vertical Height

Consider the addition of incline as an extra vertical component. However, since the tower stands independently, it just maintains the vertical height of 125 ft. So, the total vertical distance remains the height of the tower: 125 ft

Step 4: Apply the Pythagorean Theorem

To find the shortest length of the wire (the hypotenuse), use the Pythagorean theorem: \( c^2 = a^2 + b^2 \), where \( a \) is the effective horizontal distance, \( b \) is the vertical height of the tower, and \( c \) is the length of the guy wire. After computing, \( c = \sqrt{ (b)^2 + (a)^2 } \).

Step 5: Substitute Values and Calculate Wire Length

Substitute \( b = 125 \) ft and \( a = 55 \times \cos(32^{\circ}) \) into the equation from Step 4. After calculating, you will find the value of \( c \), which is the length of the wire needed.

Key Concepts

Inclined Plane

An inclined plane is a flat surface tilted at an angle relative to the horizontal. It helps us understand how different forces and components interact when they're not just moving straight up or down but at angles. In this context, the mountain side where the tower stands is the inclined plane.
It's inclined at an angle of 32° to the horizontal. This angle affects how we calculate distances since the ground itself is not flat. When a problem involves an inclined plane, it often means we need to consider how much of a distance is "horizontal" versus how much is at an angle.
In this problem, understanding the inclination helps us break down the position of the tower and the attached guy wire. Only then can we properly apply the mathematical tools needed to solve for the shortest length of this wire.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used in geometry, particularly when dealing with right triangles. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Mathematically, it's written as \( c^2 = a^2 + b^2 \). This becomes essential when trying to find unknown distances when given a couple of known ones in the context of right triangles. In the tower problem, the task is to find the length of the hypotenuse, which is the guy wire.
- Here, "a" represents the effective horizontal distance (adjusted for the inclination).
- "b" is the tower's vertical height, 125 ft.
By substituting these values into the theorem, you can find the correct wire length needed.

Horizontal and Vertical Components

When dealing with angled or inclined scenarios, breaking a force or distance into horizontal and vertical components is incredibly helpful. This simplifies the problem by addressing each component separately.
In the case of the tower, the original horizontal distance is given as 55 ft downhill, but because the mountain is inclined at 32°, this affects how far horizontally that point actually is. By using the cosine function, we can determine the effective horizontal distance as \( 55 \times \cos(32^{\circ}) \).
The vertical component remains straightforward in this scenario, representing the direct height of the tower, 125 ft.
When the horizontal and vertical distances are known, these can be pieced together in calculations, such as with the Pythagorean theorem, to find additional unknowns like the guy wire length, effectively showing how essential it is to clearly understand each component when solving trigonometric problems related to inclined planes.