00:01
Hello, students.
00:02
Today we're looking at a related rates problem.
00:06
And with related rates problems, i really like to look at a picture, and then i like to write down what i know about that picture and that scenario.
00:20
And then i'd like to write out what i'm trying to find.
00:23
So we have a spherical balloon with some radius.
00:33
And you can call that radius r.
00:43
And we know that the radius at a particular moment in time is we have that it's 30 and we also have later on that it is 60 so these are our two questions that are being asked is something about the radius at 30 degrees 30 centimeters and 60 centimeters we also know that the volume of the blue is increasing so the balloon as gas gets pumped in the balloon is expanding in all directions at a rate of 800 cubic centimeters per minute well that's the rate of the volume so i'm gonna go ahead and call that dv d t and then the last thing i do know is that we have a volume formula for a sphere which is four -thirds pi are cubed.
02:23
What am i trying to find? in this one i'm trying to find what is the rate at which the radius is increasing when the radius is 30 centimeters and again when it's 60 centimeters.
02:46
So i'm trying to find drdt.
02:53
So that brings me back to our formula.
02:57
So we know the volume is equal to four thirds pi are cubed.
03:01
So the first thing i do with a relay rate's problem when i'm ready to start solving is i try to determine do i have any constant values that i can plug in so i don't even know what the volume is i can figure it out for 30 and 60 but it's not important in this problem because the volume is going to be changing and i know that r the radius will also be changing because i have two separate values so i know that that's not a constant pi is a constant, four thirds is a constant, but those are already plugged in.
03:37
This becomes important in problems like when i have a triangle and i know the area of a triangle is equal to one half base times height and say i want to know what rate is the height increasing when at a particular moment in time and knowing full well that the base is always five meters.
04:19
So here the base would always be five meters, but our height would be changing.
04:25
And so we would actually be able to plug in five before we take the derivative.
04:32
With this problem, since we do not have any constants, we do not have any values that would be staying the same for volume or radius.
04:42
We're not going to do that.
04:43
So the first thing i'm going to do is i'm going to take the derivative dv d t.
04:54
And then i take the derivative of four -thirds pi r cubed.
05:01
Well, that's going to be three using the power rule for r cubed, because we're taking this with respect two times.
05:09
So that's why you have dv d t.
05:11
And we take the derivative of r.
05:14
So that would be three times four -thirds.
05:18
Sorry, we're taking the derivative of r cubed.
05:21
So 3 times 4 thirds, pi r squared times d r d t.
05:40
Well, the 3 is cancel, and you're left with 4 pi r squared dr d t.
05:57
So then i'm going to come over here, and this is the part at which i plug in my values.
06:05
So i have essentially 4 pi r squared d rd t is equal to dvd t.
06:11
What's really cool is if you notice 4 pi r squared, d rd t, is equal to dvd t.
06:15
Squared that's the surface area of a sphere and so when you take the derivative of the volume we get the surface area for a sphere i always thought that was kind of cool all right so here i'm gonna plug in 800 for dvdt so i'll have 800 is equal to 4 pi times i'm gonna do a first so times 30 squared and i'm just going to write r prime.
07:08
And remember r prime and drdt, these are the same thing...