Write the distance y across the flat sides of a hexagonal nut as a function of r (see figure).
www.webassign.net/larpcalclim2/4-8-049.gif
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Yeah
In the given figure, the distance y across the flat sides of a hexagonal nut can be calculated as a function of r.
First, let's divide the hexagonal nut into six equilateral triangles. Each equilateral triangle has sides of length r.
Now, we need to find the height of each equilateral triangle. The height can be calculated using the formula:
height = (sqrt(3) / 2) * side
In this case, the side length is r, so the height of each equilateral triangle is:
height = (sqrt(3) / 2) * r
Since the hexagonal nut has six equilateral triangles, there are six heights. The total distance y across the flat sides of the hexagonal nut is equal to the sum of all the heights:
y = 6 * [(sqrt(3) / 2) * r]
Simplifying the expression, we get:
y = 3 * sqrt(3) * r
To find the distance y across the flat sides of a hexagonal nut as a function of r, we can use the formula for the apothem of a regular hexagon.
The apothem of a regular hexagon is the distance from the center of the hexagon to the midpoint of a side. In this case, the distance y across the flat sides of the nut is equivalent to twice the apothem.
The formula for the apothem of a regular hexagon is given by:
apothem = r * √3
where r is the radius of the circumscribed circle.
Therefore, the distance y can be expressed as:
y = 2 * apothem
= 2 * r * √3
So, the function for y in terms of r is:
y = 2r√3
It will be twice the height of one of the equilateral triangles.
Concentrate on one of these ...
Construct the height h, you now have a right-angled triangle with angles 30-60-90°
such a triangle has sides in the ratio of 1:√3:2
so setting up a proportion:
(r/2) : 1 = h : √3
h = √3 r/2
you need 2h