Write the distance y across the flat sides of a hexagonal nut as a function of r.
divide the nut into 6 equilateral triangles, with sides = r.
y=r
To determine the distance y across the flat sides of a hexagonal nut as a function of r, we need to first understand the geometry of a regular hexagon.
A regular hexagon consists of six equal sides and six equal angles. If we draw two lines from opposite corners of the hexagon, it divides the hexagon into two equilateral triangles.
Now, let's consider a regular hexagon nut. The distance across the flat sides, y, can be determined by finding the height of one of these equilateral triangles.
To find the height, we can use the Pythagorean theorem. The height (h) is the distance from the center of the hexagon to one of the vertices, which is also the radius (r). The base (b) of the equilateral triangle is one of the sides of the hexagon.
Applying the Pythagorean theorem:
h^2 = r^2 - (b/2)^2
In an equilateral triangle, the ratio of the height to the base is √3/2.
Therefore, b = 2r and h = (√3/2) * b.
Substituting the values of b and h, we get:
h^2 = r^2 - (2r/2)^2
h^2 = r^2 - r^2
h^2 = 0
Hence, the height of the equilateral triangle formed by the flat sides of the hexagonal nut is zero. Therefore, the distance y across the flat sides is also equal to zero.
In conclusion, the distance y across the flat sides of a hexagonal nut, as a function of the radius r, is y(r) = 0.