00:01
So in this problem, we're given a bowl that is 30 centimeters in diameter, and we drop a 10 centimeter diameter ball in this bowl, a heavy ball, and we begin to fill it with water.
00:18
And the water level in here, right, is at some height h.
00:29
Okay, and the question is, for us to figure out the formula, find the volume of water in the bowl based on that depth h.
00:42
Okay.
00:44
So first of all, we know that for the bowl, the diameter is 30 centimeters, so therefore the radius is 15 centimeters.
01:02
Okay.
01:04
Let's assume that we do this, right? we have the origin be at the very bottom of the bowl here.
01:12
And the x and y axis go through the middle of it just like that drawing right there okay then using the center radius form of a circle this this equation of this bulk is the hemisphere so it's half a circle right is x squared plus y minus 15 squared i guess i should have put my x -axis up here at the top 15 squared is equal to 15 squared okay so as i've said we'll put our x -axis up here okay now let's calculate this for values of x then right so this is x squared plus y squared actually we'll just say that x squared equal 225 that's 15 squared minus y minus 15 squared and so then x is 225 minus 15 squared a square and this is going to be for zero it's going to be valid for zero is less than or equal to y is less than or equal to 15 now which matches is we can't have a height greater than 15 centimeters because our mole's only 15 centimeters, right? so then the volume is going to be found by rotating these values, rotating this with respect to the y -axis, isn't it? just rotate those y -values around the x -values around the y -axis.
04:03
Okay.
04:05
So that means i end up with pi times the integral from 0 to h of the square root of 225 minus y minus 15 squared.
04:34
And you square this, pi r squared, d y, right? okay.
04:42
So that means then that i get pi times the integral from 0 to h of 225 minus y squared minus plus 30y minus 205 minus 225 minus 225.
05:14
All of this, d .y.
05:18
Well, 225 minus 225 cancels out, doesn't it? okay.
05:25
So that means i have pi times, let's see.
05:34
The integral of 30y is 30 y over 2 and that's y squared minus y cubed over 3 we're going to evaluate this from 0 to h okay well when i put in zero in there for y every time that's all zero so that's all going to can't drop out and so that means i'm left with 15 h squared let me do it this way.
06:20
Let me do it this way and make it clearer.
06:21
Pi times 15 h squared minus h cubed over 3.
06:35
Okay.
06:36
And of course, this is from zero is less than equal to h is less than equal to 15.
06:42
So this is the volume in the bowl.
06:49
All right.
06:50
So similarly, we do the same thing for the ball, right? for the ball, we have diameter is 10 centimeters, and so the radius is 5 centimeters.
07:15
Again, doing the same thing like we did earlier.
07:19
We have the ball here, our x and y this way...