Solving for NL
This task requires the use of basic trigonometric concepts. Let us first draw a diagram to visualize the problem:
In the diagram, we have a point M at a distance m from the plane, and two inclined lines MN and ML at angles of 30° and 60° respectively. We are asked to find the length of line NL, which forms the hypotenuse of a right triangle formed by MN and ML.
Since the angles of 30° and 60° form a 90° angle with the plane, we can use the trigonometric ratios of sine and cosine to solve for NL. Let's start by finding the lengths of MN and ML.
MN = m sin(30°)
= m * 1/2
= m/2
Similarly, ML = m sin(60°)
= m * √(3)/2
= √(3)m/2
Using the Pythagorean theorem, we can find the length of NL:
NL² = (m/2)² + (√(3)m/2)²
= m²/4 + 3m²/4
= 4m²/4
= m²
Taking the square root, we get:
NL = √(m²)
= m
However, this is the length of NL projected onto the plane. To find the actual length, we need to find the perpendicular distance from M to NL, which is the height of the triangle formed by MN and NL. Since we know the angles of 30° and 60°, we can use the tangent ratio:
tan(30°) = Height/NL
Height = NL*tan(30°)
= m*tan(30°)
Similarly,
tan(60°) = Height/NL
Height = NL*tan(60°)
= m*tan(60°)
Since tan(60°) = √(3), we get:
Height = √(3)m/3
Therefore,
NL = √(m² + (√(3)m/3)²)
= √(m² + 3m²/9)
= √(4m²/9)
= 2m/√(3)
So, we have finally found the length of NL to be 2m/√(3), or simply, 2a/√(3) (since NL = 2a).
Therefore, NL = 2a/√(3).
The explanation for this solution is that the angle of 60° is double the angle of 30°, and since they both form a 90° angle, the length of NL can be found by using the sine and cosine ratios of a right triangle. Since we know that MN and NL are perpendicular to each other, we can use the Pythagorean theorem to find the length of NL. Additionally, the perpendicular distance from M to NL can also be found by using the tangent ratio.
Disclaimer: This task is purely for academic purposes and should not be used for any other purposes. Cheating and violating academic integrity is a serious offense and can lead to severe consequences.
