Answer:
Explanation:
I had no idea what a frustum was until now. neat.
Looks like we want to find the volume of the whole thing and then subtract the volume of the tip hat gets cut off, so let's find both those.
Volume of a pyramid is area of the base times its vertical, or perpendicular height divided by 3, or put into variables (l*w*h)/3 with l*w being the area of the base.
To find the length and width of the base, since both are the same because it's a square base, we can form a right triangle with the perpendicular height and the side length. The side length is the hypotenuse and the height is, well, the height. Finding the third side with the Pythagorean theorem will give us the distance from the center of the square to one of its sides. So to find the length or width we are going to want to double it.
So let's get to the Pythagorean theorem.
a^2 + b^2 = c^2
a^2 + 180^2 = 200^2
a^2 = 7600
a = sqrt(76) gonna keep it like that to minimize rounding. And remember we want twice that for the lengths and width so the length and width is 2sqrt(7600)
So now we can find the volume of the big pyramid with the (l*w*h)/3
(l*w*h)/3
(2sqrt(7600)*2sqrt(7600)*180)/3 (2sqrt(7600)*2sqrt(7600) is saying (2sqrt(7600))^2
((2sqrt(7600))^2*180/3)
4*7600*180/3
1,824,000 cm^3
Hopefully you know that if you cut a pyramid with a horizontal slice, the top pyramid that you take off is similar to the original. So you can find measurements with proportions. We use this to find the height of the small pyramid. If we have the proportion as small pyramid / large we get this.
h/100 = 180/200 then multiply both sides by 100
h = (180/200)100
h = 90
we can do the same for the side lengths
w/100 = 2sqrt(7600)/200
w = sqrt(7600)
Now we can find the volume of this small pyramid
(sqrt(7600*sqrt(7600)*90)/3
(7600*90)/3
338,000
And now we can solve for the volume of the frustum, if you remember it's the volume of the whole pyramid minus the volume of the small one.
1,824,000 - 338,000 = 1,486,000 cm^3
Let me know if there is something you don't understand.