a tree on a 30 degrees grows straight up. What are the measures of the greatest and smallest angles the tree makes with the hill? explain
Hard to parse the garbled sentence, but if the hill slopes at a 30 degree angle, then the tree makes angles of
60 and 120 with the hill.
To explain, just draw a diagram and label all the angles.
Well, well, well, let me branch out my knowledge here. So, the tree is growing straight up on a 30-degree hill, huh? That's quite the incline for a tree. Now, let's get down to the angles.
The greatest angle the tree makes with the hill would be 90 degrees. Why, you ask? Well, think about it. If the tree is growing straight up, perpendicular to the ground, it forms a right angle with the hill. So, 90 degrees is the way to go.
Now, for the smallest angle. Picture this: If the tree is growing straight up, but the hill is tilting at 30 degrees, there will be a bit of a slope. So, the smallest angle would be the angle between the tree and the slope of the hill. This angle is like the tree's "lean" against the hill. So, it would be 30 degrees. That's the littlest angle you'll find here.
Hope that explains it in a way that leaves you feeling satisfied!
To find the measures of the greatest and smallest angles the tree makes with the hill, we need to consider the properties of triangles and trigonometry. Let's break down the problem step by step:
Step 1: Draw a diagram
Start by drawing a horizontal line to represent the hill and a vertical line to represent the tree. Label the angle between the tree and the hill as ∠A.
A
/|
tree / |
/ |
/θ |
/ | hill
--------
β
Step 2: Identify angles and trigonometric functions
In this problem, we know that the angle between the tree and the hill is 30 degrees (∠A = 30°). We need to find the measures of the other two angles (∠β and ∠θ).
Step 3: Use the properties of a triangle
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write:
∠A + ∠β + ∠θ = 180°
Step 4: Find the measure of ∠β
From the given information, we know that ∠A = 30°. By substituting this value into the equation from step 3, we get:
30° + ∠β + ∠θ = 180°
Simplifying the equation gives:
∠β + ∠θ = 150°
Step 5: Consider the properties of a right triangle
The tree is growing straight up, so it forms a right angle (∠θ) with the ground. In a right triangle, the sum of the non-right angles is always 90 degrees. Therefore, we can write:
∠θ + ∠β = 90°
Step 6: Solve the system of equations
We now have a system of two equations with two unknowns (∠β + ∠θ = 150° and ∠θ + ∠β = 90°). Solving this system of equations will give us the measures of ∠β and ∠θ.
There are several methods to solve this system, such as substitution or elimination. By adding the two equations, we have:
(∠β + ∠θ) + (∠θ + ∠β) = 150° + 90°
2(∠β + ∠θ) = 240°
∠β + ∠θ = 120°
Now we have a new equation: ∠β + ∠θ = 120°
Subtracting this equation from our previously derived equation (∠β + ∠θ = 150°), we get:
(∠β + ∠θ) - (∠β + ∠θ) = 150° - 120°
0 = 30°
This result is incorrect since it implies that 0 = 30°. Therefore, there is no valid solution for this system of equations.
To summarize:
- We found that the given information leads to an invalid system of equations.
- Therefore, it is not possible to determine the measures of the greatest and smallest angles the tree makes with the hill based on the given information.
To determine the measures of the greatest and smallest angles the tree makes with the hill, we need to consider the relationship between the tree's vertical growth and the slope of the hill.
Let's break this down step by step:
1. First, visualize the scenario. Imagine a hill with a slope at a 30-degree angle relative to the horizontal ground, and a tree growing straight up on this hill.
Now, we can observe that the tree's straight vertical growth is perpendicular to the hill's slope.
2. The greatest angle the tree makes with the hill is formed when the tree points directly against the slope, which means it aligns itself perpendicular to the slope. Since the hill has a slope angle of 30 degrees, the greatest angle the tree makes with the hill will also be 30 degrees. This angle can be seen as a right angle or 90 degrees minus the slope angle of the hill.
3. On the other hand, to determine the smallest angle the tree makes with the hill, we need to consider the overall shape of the scenario. The smallest angle would occur when the tree is parallel to the slope, meaning it follows the same direction as the hill's slope.
Since the tree grows straight up at a 90-degree angle from the ground, the smallest angle the tree makes with the hill would be 90 degrees minus the slope angle of the hill, which is 30 degrees. Therefore, the smallest angle in this case is 60 degrees.
In summary, the greatest angle the tree makes with the hill is 30 degrees, while the smallest angle is 60 degrees.