Equilateral hexagon is revolving around one of its edges. Find the volume of the solid of revolution.
no idea how to do this can someone please help!!! urgent!
One way: Theorem of Pappas
If the edge goes from (0,0) to (0,1), then the center is at (√3/2,√3/2).
The area of the hexagon is 3√3/2. The radius of rotation is √3/2, so its path has length π√3.
So, the volume of the solid is π√3 * 3√3/2 = 9π/2
Another way: calculus. Using symmetry, we can rotate a triangle and a rectangle about the y-axis.
Triangle: using discs, the volume is
v = ∫[1,3/2] π(R^2-r^2) dy
where R=(√3(3/2-y) and r=(√3(y-1/2))
v = ∫[1/2,1] π((√3(3/2-y))^2-(√3(y-1/2))^2) dy = 3π/4
Rectangle:
v = ∫[0,1/2] πr^2 dy
where r=√3
v = ∫[0,1/2] 3π dy = 3π/2
That is the volume of the top half of the figure: 9π/4
Double that and you get the first volume: 9π/2
To find the volume of the solid of revolution formed by a revolving equilateral hexagon, you can follow these steps:
1. Understand the problem:
- An equilateral hexagon is a six-sided polygon with all sides and angles equal.
- We are revolving the hexagon around one of its edges, so it forms a three-dimensional solid.
2. Determine the given measures:
- Since specific dimensions for the hexagon are not provided, assume a side length "s" for the equilateral hexagon.
- The edge being revolved will be a side of the hexagon, which has a length of "s".
3. Visualize the solid of revolution:
- When the equilateral hexagon is revolved around one of its edges, it creates a frustum of a cone.
- The base of the frustum is the original hexagon, and the top is a smaller hexagon since the edge is revolving inward.
4. Calculate the dimensions of the frustum:
- The larger base radius is equal to the distance from the center of the hexagon to one of its vertices. This can be found using the formula: r = s/√3.
- The smaller top radius is equal to the distance from the center of the hexagon to the midpoint of one of its sides. This can be found using the formula: R = s/(2√3).
- The height of the frustum is equal to the length of the edge being revolved, which is "s".
5. Calculate the volume of the frustum:
- The formula to find the volume of a frustum of a cone is: V = (1/3)πh(R^2 + Rr + r^2), where h is the height and R and r are the radii of the top and bottom bases, respectively.
- In this case, the height (h) is equal to "s", the smaller top radius (R) is s/(2√3), and the larger bottom radius (r) is s/√3.
6. Substitute the values and solve for volume:
- V = (1/3)πs(s/(2√3)^2 + s(s/(2√3) * s/√3) + (s/√3)^2)
- Simplify the equation further and calculate the value of V.
By following these steps, you should be able to find the volume of the solid of revolution of the equilateral hexagon.