The trunk of a leaning tree makes an angle of 12 degrees with the vertical. To prevent the tree from falling over, a 35.0 m rope is attached to the top of the tree and is pegged into level ground some distance away. if the tree is 20.0 m from its base to its top, calculate the angle the rope makes with the ground to the nearest degree

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To find the angle the rope makes with the ground, we can use trigonometry. Let's call this angle θ.

We have a right triangle formed by the height of the tree, the distance from the base to the point where the rope is attached, and the length of the rope.

In this case, the height of the tree is the side opposite to the angle θ, the distance from the base to the point where the rope is attached is the side adjacent to the angle θ, and the length of the rope is the hypotenuse.

We can use the tangent function to find the angle θ:

tan(θ) = opposite/adjacent

tan(θ) = height/distance

tan(θ) = 20.0 m/distance

To calculate the distance, we can use the tangent of 12 degrees:

tan(12°) = 20.0 m/distance

Now we can rearrange the equation to solve for distance:

distance = 20.0 m/tan(12°)

Using a calculator, we can evaluate tan(12°):

tan(12°) ≈ 0.2126

distance = 20.0 m/0.2126 ≈ 94.03 m

So the distance from the base to the point where the rope is attached is approximately 94.03 meters.

Finally, we can find the angle the rope makes with the ground using the inverse tangent function:

θ ≈ tan^(-1)(20.0 m/94.03 m)

θ ≈ 12.18°

Therefore, the angle the rope makes with the ground is approximately 12 degrees.

To solve this problem, we can use trigonometry. Let's define some variables:

θ: The angle the rope makes with the ground
h: The height of the tree
d: The distance from the base of the tree to the point where the rope is pegged into the ground

We know that the trunk of the leaning tree makes an angle of 12 degrees with the vertical. Since the tree is leaning, the angle between the trunk and the ground is 90 - 12 = 78 degrees.

Now, let's draw a right-angled triangle to represent the situation. The height of the tree h is the vertical side (adjacent to the angle of 78 degrees) and the distance d is the horizontal side (opposite to the angle of 78 degrees). The rope acts as the hypotenuse of the triangle.

Using trigonometry, we can relate the angles and sides of a right-angled triangle using the tangent function:

tan(78) = h / d

Rearranging the equation, we can solve for h:

h = d * tan(78)

Now, we can substitute the given values into the equation:

h = 20.0 m
θ = the angle the rope makes with the ground (which we want to find)

Since the length of the rope is 35.0 m, it is equal to the hypotenuse of the right-angled triangle:

hypotenuse = √(h^2 + d^2)

Substituting the values we know, we have:

35.0 = √((20.0)^2 + d^2)

Solving this equation for d:

d^2 = (35.0)^2 - (20.0)^2
d^2 = 1225 - 400
d^2 = 825
d = √825

Now, we can substitute the values of h and d into our initial equation to solve for θ:

h = d * tan(θ)

Rearranging the equation, we have:

tan(θ) = h / d

Substituting the values:

tan(θ) = 20.0 / √825

To find the angle θ, we can take the arctangent (inverse tangent) of both sides:

θ = arctan(20.0 / √825)

Evaluating this using a calculator, we find:

θ ≈ 63 degrees

Therefore, the angle the rope makes with the ground is approximately 63 degrees (to the nearest degree).

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