00:01
Let's first establish what type of shape this has to be.
00:04
If the cross section of this shape is an equilateral triangle, then we're looking at a cone.
00:11
So if we're dealing with volume of a cone, then in order to find that, you need the area of the base times the height divided by three because it's a third of a cylinder's volume.
00:22
Well, the base area of a cone is a circle, which is pi r squared.
00:25
So you can see up there at the top there, i have that the cone's volume is pi r squared times the height divided by three.
00:31
Three.
00:32
Because you know all three sides of the equilateral triangle, you can figure out the radius.
00:39
The radius would be two.
00:41
And the base of that triangle would be four, just like all the other sides of the equilateral triangle.
00:47
But in order to find the initial height of this cone that we're trying to find the volume of, we need to split this equilateral triangle in half in the fashion of two, 36.
01:01
6090 triangles.
01:03
And if you're familiar with 30 6090 triangles, you have a short leg, a long leg, and a hypotenuse.
01:10
In our instance, what we're working with here, the hypotenuse is the side of the cone.
01:17
And the long leg of the 306090 represents, in this case, the height of the pyramid.
01:24
I'm sorry, not pyramid cone.
01:26
And the short leg of this 306090 is represented.
01:31
By half of the base, which is the radius, which is two.
01:37
So there is a trick with special right triangles.
01:40
You could do the pythagorean theorem if you'd like, but i just like to use that i know in the 30, 60 90, the longer leg, which is the side across from the 60 degree angle is the shorter leg times the square root of three.
01:54
So i can then conclude that the height of this original cone with this cross section would be two radical three, two square root three meters.
02:04
So all that being said, i go back up to the volume calculation.
02:09
So here you can see i have pi times two squared because r squared the radius is two times the height, which we just calculated using a 3060 90, which is two radical three divided by three.
02:21
As we work through here and simplify some things, two squared is four, four times two radical three is eight radical three.
02:31
So we end up with this, i mean, it's simplified as far as we can, eight radical three times pi divided by three, which is about 14 .5 meters cubed.
02:42
And i went ahead and i wrote that approximation out to the side just so that we can check the answer at the end, but i'm not sure exactly how your teacher would like you to write your final answer.
02:53
So we're not done here...