Write the distance y across the flat sides of a hexagonal nut as a function of r (see figure in the website below).
www.webassign.net/larpcalclim2/4-8-049.gif
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Enclose the hexagon within a square
Consider the right-angled triangle at the top right.
It is a 30-60-90° triangle with a hypotenuse of r
(The sides of this triangle are in the ratio of 1 : √3 : 2 )
let the side opposite the 60° angle be p
p/r = √3/2
p = √3r/2
but y = 2p = √3r
Why did the nut become a stand-up comedian? Because it wanted to crack hilarious jokes and be the "nuttiest" of them all!
But to answer your question seriously, we cannot directly measure the distance y across the flat sides of a hexagonal nut from the given website. However, we can make an educated guess that y could be related to the radius r of the nut. Can I assist you with any other hilarious questions or thoughts?
In the given figure, the distance across the flat sides of a hexagonal nut is represented by y.
To find an expression for y in terms of r, let's analyze the figure:
The hexagonal nut can be divided into three consecutive equilateral triangles, as shown in the figure.
Each side of the equilateral triangle forms one of the six flat sides of the hexagonal nut.
Let's consider one of these equilateral triangles.
The length of each side of the equilateral triangle is equal to the radius of the hexagonal nut, which is represented by r.
In an equilateral triangle, the distance from the center to any vertex is equal to 2/3 times the height (h) of the triangle.
Since the height of the equilateral triangle is equal to the distance across the flat sides of the hexagonal nut (y), we have:
y = 2/3 * h
Now, let's find the height of the equilateral triangle.
In a right triangle formed by one side of the equilateral triangle, the height, and the hypotenuse (side of the hexagonal nut), we can apply the Pythagorean theorem.
The hypotenuse is equal to twice the radius of the hexagonal nut, so it is equal to 2r.
Using the Pythagorean theorem, we have:
(h/2)^2 + r^2 = (2r)^2
Simplifying the equation:
(h^2)/4 + r^2 = 4r^2
Multiplying both sides by 4:
h^2 + 4r^2 = 16r^2
Subtracting 4r^2 from both sides:
h^2 = 12r^2
Taking the square root of both sides:
h = √(12r^2) = √(4 * 3r^2) = 2√(3r^2) = 2r√3
Now we can substitute the value of h into the equation for y:
y = 2/3 * h = 2/3 * (2r√3) = 4r√3/3
Therefore, the expression for the distance across the flat sides of a hexagonal nut, y, in terms of the radius of the nut, r, is:
y = (4r√3)/3
To determine the distance y across the flat sides of a hexagonal nut as a function of r (the radius), we need to use some properties of a regular hexagon.
In a regular hexagon, all sides are equal in length, and the opposite sides are parallel. The distance across the flat sides, y, can be found by considering the relationship between the radius and the side length of the hexagon.
First, let's consider the smaller equilateral triangle formed by two radii and one side of the nut. This triangle has three equal sides. The central angle of the triangle is 60 degrees since a hexagon has six equal angles.
Now, using trigonometry, we can find the length of each side of the equilateral triangle in terms of the radius r. The side length of the triangle is equal to 2r * sin(30 degrees) since the central angle is 60 degrees.
sin(30 degrees) = 1/2, so the side length of the equilateral triangle is equal to r.
Since the distance y across the flat sides of the hexagonal nut is twice the side length of the equilateral triangle, we have:
y = 2 * r
Therefore, the distance y across the flat sides of a hexagonal nut as a function of r is simply:
y = 2r