A 12-foot ladder is leaning against the outside wall of a building. The angle of the ladder with the ground is 75 degrees.
The ladder is moved so that the base of the ladder is 5.8 feet from the wall. Determine the angle created by the ladder and the ground after the move. Round your answer to the nearest tenth of a degree.
if the angle is x, then
5.8/12 = cos x
so, x = 61.1°
To determine the angle created by the ladder and the ground after it is moved, we can use trigonometric ratios.
Let's define some variables:
- Let θ be the angle between the ladder and the ground after it is moved.
- Let a be the distance between the base of the ladder and the wall after it is moved.
First, let's find the length of the ladder along the ground after it is moved. Using the Pythagorean theorem, we have:
c^2 = a^2 + b^2,
where c represents the length of the ladder (which remains 12 feet), and b represents the height of the ladder along the wall (which remains the same).
Substituting the given values, we have:
12^2 = a^2 + 5.8^2.
Simplifying:
144 = a^2 + 33.64,
a^2 = 144 - 33.64,
a^2 = 110.36,
a ≈ √110.36,
a ≈ 10.5 feet.
Now that we have determined the distance along the ground, we can use the tangent function to find the angle θ. In this case, the tangent function is given by:
tan(θ) = b/a.
Substituting the known values, we have:
tan(θ) = 5.8/10.5.
Using a calculator, we can solve for θ:
θ ≈ arctan(5.8/10.5).
Evaluating this expression, we find that:
θ ≈ 0.504 radians.
To convert this into degrees, we multiply by 180/π:
θ ≈ 0.504 * (180/π) ≈ 28.9 degrees.
Therefore, the angle created by the ladder and the ground after it is moved will be approximately 28.9 degrees, rounded to the nearest tenth of a degree.
To determine the new angle created by the ladder and the ground after it is moved, we can use the tangent function.
Given:
Length of the ladder (hypotenuse): 12 ft
Original angle with the ground: 75 degrees
Distance of the base from the wall: 5.8 ft
Since we have the opposite (base) and hypotenuse, we can use the formula for tangent:
tan(angle) = opposite / adjacent
Let's calculate the original opposite (base) and adjacent (height of the building) using trigonometric ratios:
Opposite = length of the ladder * sin(angle)
Adjacent = length of the ladder * cos(angle)
Opposite = 12 ft * sin(75 degrees)
Opposite ≈ 11.311 ft
Adjacent = 12 ft * cos(75 degrees)
Adjacent ≈ 3.073 ft
Now, let's calculate the new angle using the tangent function:
tan(angle) = opposite / adjacent
angle = arctan(opposite / adjacent)
angle = arctan(11.311 ft / 5.8 ft)
angle ≈ 63.7 degrees
Therefore, after the ladder is moved, the new angle created between the ladder and the ground is approximately 63.7 degrees.