Angle Of Depression: Car To 150m Tower

by Jhon Lennon 39 views

Hey guys, let's dive into the fascinating world of trigonometry and tackle a classic problem: the angle of depression! We're going to figure out how to calculate the angle of depression of a car parked on the road from the top of a 150-meter-high tower. This isn't just some abstract math problem; understanding these concepts can be super useful in real-world scenarios, like navigation, surveying, and even when you're trying to estimate distances from a high vantage point. So, buckle up, and let's break down this problem step-by-step.

Understanding the Angle of Depression

First things first, what exactly is the angle of depression? Imagine you're standing at the top of a tall building, like our 150-meter tower, and you're looking down at an object on the ground, like a car. The angle of depression is the angle formed between your horizontal line of sight and the line of sight down to the object. It's crucial to remember that this angle is always measured downward from the horizontal. Think of it this way: if you were to extend your line of sight straight forward horizontally, the angle of depression is how much you have to tilt your head down to see the car. It's important not to confuse this with the angle of elevation, which is the angle measured upward from the horizontal.

Now, let's visualize our scenario. We have a tower that's 150 meters tall. At the very top of this tower, someone is looking down at a car parked on the road below. We want to find the angle of depression. To do this mathematically, we'll often use a diagram. We can represent the tower as a vertical line, the ground as a horizontal line, and the car as a point on that horizontal line. The line of sight from the top of the tower to the car forms the hypotenuse of a right-angled triangle. The height of the tower is one of the legs (the opposite side to the angle we're interested in if we consider the angle from the car's perspective), and the horizontal distance from the base of the tower to the car is the other leg (the adjacent side).

Here's where a cool trick comes in handy: the angle of depression from the top of the tower to the car is equal to the angle of elevation from the car to the top of the tower. Why? Because the horizontal line of sight from the top of the tower is parallel to the ground. When a transversal (the line of sight to the car) intersects two parallel lines, the alternate interior angles are equal. This means we can often solve for the angle of depression by considering the angle of elevation from the object on the ground, which can sometimes simplify the diagram and calculations. So, while we're talking about the angle of depression from the tower, we can use the properties of parallel lines to relate it to an angle of elevation from the car.

Let's solidify this with our specific example. We have a tower of height h=150h = 150 m. Let the car be parked at a distance dd meters from the base of the tower. Let θ\theta be the angle of depression from the top of the tower to the car. If we draw our diagram, we'll see a right-angled triangle. The vertical side is the height of the tower (150 m). The horizontal side is the distance from the base of the tower to the car (dd). The line of sight is the hypotenuse. The angle of depression, θ\theta, is outside this triangle, measured down from the horizontal. However, the angle of elevation from the car up to the top of the tower is inside the triangle, at the car's position, and this angle is also θ\theta. This is the key insight that makes solving these problems much more straightforward. We can use trigonometric ratios (SOH CAH TOA) with this right-angled triangle.

So, to actually calculate the angle of depression, we need one more piece of information: the horizontal distance from the base of the tower to the car. Without that distance, we can't find a specific numerical value for the angle. However, we can set up the equation. If we know the distance dd, we can use the tangent function. Remember TOA? Tangent is Opposite over Adjacent. In our right-angled triangle, the side opposite the angle θ\theta (at the car) is the height of the tower (150 m), and the side adjacent to θ\theta is the distance dd. Therefore, we have: tan⁡(θ)=oppositeadjacent=150d\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{150}{d}. To find the angle θ\theta, we would then use the inverse tangent function (arctan or tan⁻¹): θ=arctan⁡(150d)\theta = \arctan(\frac{150}{d}). This formula is our gateway to finding the angle of depression once we have the distance dd. It's pretty neat how we can use simple geometry and trigonometry to solve what might seem like a complex observational problem. Stay tuned as we work through an example!

Calculating the Angle with Distance

Alright guys, now that we've got a solid grasp of what the angle of depression is and how it relates to the angle of elevation, let's put it into practice! To actually calculate the angle of depression of the car from the top of our 150-meter tower, we need one crucial piece of information: the horizontal distance from the base of the tower to the car. Without this distance, we can express the angle in terms of variables, but we can't get a concrete number. Let's assume, for the sake of this example, that the car is parked 200 meters away from the base of the tower. So, our height h=150h = 150 m and our distance d=200d = 200 m.

Remember our diagram? We have a right-angled triangle. The vertical side is the height of the tower (150 m), and the horizontal side is the distance from the base of the tower to the car (200 m). The angle of elevation from the car to the top of the tower is the angle inside the triangle at the car's position. As we discussed, this angle is equal to the angle of depression from the top of the tower. Let's call this angle θ\theta. We want to find θ\theta.

We need to pick the right trigonometric ratio. We have the side opposite to the angle θ\theta (the height of the tower, 150 m) and the side adjacent to the angle θ\theta (the distance from the base, 200 m). The trigonometric ratio that relates the opposite and adjacent sides is the tangent. So, we use the formula: tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}.

Plugging in our values, we get:

tan⁡(θ)=150 m200 m\tan(\theta) = \frac{150 \text{ m}}{200 \text{ m}}

tan⁡(θ)=1520\tan(\theta) = \frac{15}{20}

tan⁡(θ)=34\tan(\theta) = \frac{3}{4}

tan⁡(θ)=0.75\tan(\theta) = 0.75

Now, to find the angle θ\theta, we need to use the inverse tangent function, also known as arctangent (often written as arctan⁡\arctan or tan⁡−1\tan^{-1}). This function essentially