A cell phone tower is anchored by two cables on each side for support. The cables stretch from the top of the tower to the ground, with each being equidistant from the base of the tower. The angle of depression from the top of the tower to the point in which the cable reaches the ground is 23°. If the tower is 140 feet tall, find the ground distance between the cables.
The Correct Answer and Explanation is:
To solve this problem, we will model it as a right triangle problem using trigonometry.
Step-by-Step Solution:
The angle of depression from the top of the tower to the ground where each cable touches is 23°, and the height of the tower is 140 feet. Since the cables are symmetrical and equidistant from the base, we can focus on one side and later double the base distance to find the total ground distance between both cables.
We are looking for the horizontal distance from the base of the tower to where the cable touches the ground (let’s call this xx). This forms a right triangle with:
- Opposite side = height of the tower = 140 ft
- Angle = 23°
- Adjacent side = xx
- Use: tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} tan(23°)=140x\tan(23°) = \frac{140}{x} Solving for xx: x=140tan(23°)≈1400.4245≈329.8 feetx = \frac{140}{\tan(23°)} \approx \frac{140}{0.4245} \approx 329.8 \text{ feet}
Since there is one cable on each side of the tower and they are symmetrical, the total ground distance between the cables is: 2x=2×329.8≈659.6 feet2x = 2 \times 329.8 \approx \boxed{659.6 \text{ feet}}
Answer: 659.6 feet\boxed{659.6 \text{ feet}}
Explanation
This problem involves the use of right triangle trigonometry and some understanding of geometry and physical layout. We are told that a 140-foot tall cell phone tower is stabilized by cables on each side. These cables are not just randomly attached—they are symmetrical and positioned such that each makes an angle of depression of 23° from the top of the tower down to where it hits the ground.
Visually, this forms two identical right triangles—one on each side of the tower—with the tower acting as the vertical side (opposite), the ground distance from the base to where the cable hits as the adjacent side, and the cable itself as the hypotenuse.
Because we know the angle and the vertical height, we can use the tangent function, which relates the opposite side to the adjacent side in a right triangle: tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
Plugging in the values: tan(23°)=140x\tan(23°) = \frac{140}{x}
Solving this equation gives us the horizontal distance from the base of the tower to the point where the cable meets the ground, which is around 329.8 feet. Since there are two cables on opposite sides, the total distance across the ground from where one cable touches to the other is twice that value, or 659.6 feet.
This method not only uses trigonometry but also demonstrates how symmetry and understanding of geometric shapes help solve real-world engineering problems involving forces and balance in structures like towers.
