Math Help - Volume of 3-d shape

1. Volume of 3-d shape

The shape alone is for some reason hard for me to visualize and I also need to find the volume of the following shape called S

The base of S is the triangular region on a graph with vertices (0,0), (1,0), and (0,1). Cross sections perpendicular to the y axis are equilateral triangles.

The best thing I can come up with is some kid of skewed pyramid, and I'm clueless on the volume.

2. Originally Posted by slacker142
The shape alone is for some reason hard for me to visualize and I also need to find the volume of the following shape called S

The base of S is the triangular region on a graph with vertices (0,0), (1,0), and (0,1). Cross sections perpendicular to the y axis are equilateral triangles.

The best thing I can come up with is some kid of skewed pyramid, and I'm clueless on the volume.
Note what you are trying to do here. You are finding the volume of the solid via the method of slicing:

First plot the triangular region. Once you do that, you can find the equation of the line that is created, connecting the points (0,1) and (1,0). I leave it for you to show that the line is $y=1-x$.

Now, what do we do about each individual cross section?

Recall that the area of a triangle is $A=\tfrac{1}{2}bh$

The base here, is the distance from our line to the x axis: $b(x)=1-x$

I leave it for you to show that the height of the equilateral triangle is $\frac{\sqrt{3}}{2}b(x)=\frac{\sqrt{3}}{2}(1-x)$

So, $A=\frac{\sqrt{3}}{4}(1-x)^2$

Thus, $V=\frac{\sqrt{3}}{4}\int_0^1 (1-x)^2\,dx$

Can you try to make sense out of what I did?

--Chris