00:01
Okay, so in this problem, we have a conical reservoir with the tip pointing down.
00:09
And we're told that the base radius, so this distance to there, that's 45 meters, and this height from there to there, is 6 meters.
00:24
So obviously, this is not drawn to scale.
00:26
But that's just kind of the idea.
00:29
And we're told that dvdt, since it's draining, dvdt is going to be a negative 50 cubic meters per minute.
00:41
And we want to know how fast the water is falling when the water is five meters deep.
00:47
So essentially we want to know what dhdt is.
00:52
So we want to find dhdt.
00:58
Well, to do you, that we need to get a way to relate the 2h so what we're going to do is we're going to look at a cross section of this cone as a triangle and if we if we were to come up a certain height age well then we we ought to be able to figure out what the radius is across at that distance age because it's actually going to be a similar triangle with the whole one.
01:38
So this piece right here is 45.
01:42
This is r.
01:44
This whole piece is 6, and this is h.
01:48
So we have that h over r equals 6 over 45, or 45h equals 45h equals, 6 r.
02:09
Now that means that in the volume equation where volume equals one -third times pi r squared h, we can substitute r for h.
02:28
And so we can change this to be, if i divide a 3 out on each side, i'm going to get that 15 over 2.
02:38
H equals r so that means that volume equals one third times pi times r squared so that's going to be 15 halves h squared times h okay now we can simplify that a little bit so we simplify that a little bit with algebra and we're going to to get v equals.
03:16
Well, let's see, we're going to have an h cubed, and we're going to have a 225 over 12, which will simplify that even more, pi h cubed, which, let's see, we can take three out of each of these and now give us 75 over 4 pi h cubed.
03:52
That means that dv d t equals 75 4 pi times 3h squared times d h d t.
04:18
Okay, so i've got us a little more room here.
04:20
So this is the equation that relates dvd to dhd t and we know that we are interested in it when h is 5 and we know that dvd t equals negative 50 so let's just plug those values in and we're going to have negative 50 equals 75 fourths pi times this three in there we're going to get 225 fourth pi times h squared so times 5 squared times d h d t so let's let's rearrange that a little bit solve that we're going to get that d h d t equals it equals approximately negative 0 .0113 meters per minute.
05:32
Now we could change that into different unit, easy enough...