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Show all of your work clearly:[15] Suppose that a spherical balloon is leaking: At a certain moment; the radius of the balloon is 5 cm and the balloon's volume...

Question

Show all of your work clearly:[15] Suppose that a spherical balloon is leaking: At a certain moment; the radius of the balloon is 5 cm and the balloon's volume is decreasing at 500 cm: per minute. How fast is the radius changing when the radius is 5 cm? Recall that the volume V and radius ofa sphere are related by V Tr 3

Show all of your work clearly: [15] Suppose that a spherical balloon is leaking: At a certain moment; the radius of the balloon is 5 cm and the balloon's volume is decreasing at 500 cm: per minute. How fast is the radius changing when the radius is 5 cm? Recall that the volume V and radius ofa sphere are related by V Tr 3



Answers

The radius of a spherical balloon is increasing at the rate of 3 centimeters per minute. How fast is the volume changing when the radius is 10 centimeters?

For this problem we are told that the radius of a spherical balloon is increasing at the rate of three centimeters per minutes. We have Our Prime Equals three. We are then asked how fast is the volume changing? When the radius is 10 centimeters? So we're given capital R equals 10 and we are told to use V equals for over three pi r cubed occupies approximately 3.14. So since we're asked about a rate of change, we take the derivative with respect to T implicitly. So we are implicitly assuming that are as a function of T taking the derivative. V just gives us V prime. Taking the derivative of R cubed would give us three R squared, which will divide out with the three in the denominator outfront. They're giving us four pi r squared as RV prime. Or excuse me, four pi r squared our prime as RV prime because we are essentially using chain rule there. So that will be four times 3.14 times 10 squared. So times 100 times are prime. So times three I'm going to pause and calculate that off screen. So the final result there is going to be 3768

And this problem. We're dealing with a spear whose volume is increasing at the rate of 15 inches cubed per minute. So DVD T is equal to 15 inches Q per minute. What we're tryingto find is how fast is the radius changing At the point when R is equal Teoh 10 inches. So our formula for the volume of a sphere is B equals 4/3 pi r cubed. We're gonna differentiate this with respect to time. So our DVD t is equal to four pi r squared de already t plugging in what I know DVD t is 15 I have four pi r is 10 and DRD tea is what I'm solving for So here I have 15 is equal to 400 pie Do you already t and then d already t When I divide out 400 pie is equal to 15 over 400 pie which is approximately equal 2.12 inches Her minute

For this problem. We are examining a sphere, so let's begin by drawing a picture. This is a three dimensional sphere. What do we know about this fear? Well, a sphere has a radius. Now, if the radius is constant, we could put a number on here. If the radius is changing, we're going to need to put a variable fee only numbers on our picture, our numbers that air unchanging. And we can see that the radius is of the sphere is expanding. In fact, it says it's expanding at a rate of 30 centimeters per minute. So since it's expanding, we're going to call it a variable. We'll call it our and I can write down that my radius is changing with respect to time. It's changing at a rate of 30 centimeters per minute. Okay? And we're looking at the volume. I want to know how the volume is changing with respect to time. That means what I'm looking for. It's the change of volume with respect to time. This is what I'm trying to find. So how can we relate the volume to our radius? Well, the volume of a sphere is four thirds pi are cute. So in order to find the change of volume, I need to take the derivative with respect to time. So on the left hand side, taking the derivative of V gives me d v d t On the right hand side I will take the derivative again multiplied by three four pi r That three comes down to a two and then I have to multiply this by the derivative of our with respect to t. So that's how all of these go together. Well, for this particular case, I know d r d t. That's 30. I know I want to find this. So my last piece is what is the radius? And if you read the problem, we're told that the radius at the point of time we're interested in is 15 centimeters. So I can plug in all of my values What I want to find d v d. T equals four times pi times 15 times the change of my radius, which is 30 centimeters per minute. And if I multiply all of that out, that gives me a value of before I do that. I realized I left off my squared, so it's four pi r squared d R D t So four pi times 15 squared times 30. That gives me 27,000 times pi. Or if I put that in a calculator, it is approximately 84,000 823 0.6. So that is how my volume is changing with respect to time. And if I want my units, I am in centimeters, so volume will be Centimeters Cube and my time is per minute so this will be centimeters cube per minute.

Okay, so this problems dealing with volume of a sphere. Okay, so we have a sphere and we need thio kind of Remember our volume formula. So V is equal to four thirds I r cubed. Okay, so we're dealing with the rate of change of volume. So we're gonna find the change in volume with respect to time, So formula. So we got Devi over DE T. We gotta take the respect to time on the right side. It's gonna bring the three down which will cancel with the three and the four thirds leaving us with four pi r squared times d r d t in the parliament tells us that ah are changing volume over time is equal to 15. What's the units on that cubic entrance per minute? And it wants to know d r d t When the radius is 10 so are is equal to 10 inches. So now it's just a matter of plugging in. So we have 15 is equal to four pi times 10 squared, which is 100 times d r d t solving for, um, our radius is rate of change. We just divide both sides by 400 pie s so we can get DRD T, This is our answer is it was a 0.0 119 inches per minute. It will be increasing by. Okay, thank you very much.


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