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That its volume increases at rate of 2Ocm Is_ How fast E point) Air is being pumped into spherical balloon 50 ball of radius has volume V 3Kr and IScm? Recal that s...

Question

That its volume increases at rate of 2Ocm Is_ How fast E point) Air is being pumped into spherical balloon 50 ball of radius has volume V 3Kr and IScm? Recal that surface area of the balloon increasing when its radius surtaco area $ 4xr"

that its volume increases at rate of 2Ocm Is_ How fast E point) Air is being pumped into spherical balloon 50 ball of radius has volume V 3Kr and IScm? Recal that surface area of the balloon increasing when its radius surtaco area $ 4xr"



Answers

Air is being pumped into a spherical balloon. The volume of the balloon is increasing at a rate of $20 \mathrm{cm}^{3} / \mathrm{s}$ when the radius is $30 \mathrm{cm} .$ How fast is the radius increasing at that time? (The volume of a ball of radius $r$ units is $V=\frac{4}{3} \pi r^{3}$ cubic units.)

So section 3.8 Problem number 31. Everything here is related rates. So the radius of an inflating balloon. So a spherical spherical balloon okay, is inflated at the rate of 100 pie cubic feet per minute. Okay, so that is the volume. So I know that the volume when I inflate a balloon is increasing and they're telling me that it increases at the rate of ah 100 pie. Cubic feet, huh? A minute. How fast is the radius increasing? Well, the radius would be increasing at this rate. How fast will this happen when our is equal to five. So we have to go back to our formula for the volume of a sphere is 4/3 times pi r cubed. So if we differentiate all of this with respect to T, then that tells me that d v d t is equal to this is going to give me four pi r squared d r d t So this let's be solved for d a r d t so d r d t is going to be equal to one over four pi r squared times DV the tea and so I need to evaluate this under the condition are is equal to five and DVD T is equal to 100 pie cubic feet per minute. So when I substitute all of those values in I get one over four pi r squared is going to be 25 all of this times 100 high. And so you get nice cancellations here. Pi over pies 1 100 over 100 is one. So this answer is one. That's how much the radius is creasing. So it's one feet per minute. That is D R D t. That is how the radius of this sphere is increasing. Now they ask us, how was the surface area changing? So same problem surface area of the sphere is four pi r squared. So how is that changing? Well, D S d t is going to be eight. Hi are D R d t. And so if I evaluate this when what are is equal to five, then I'm gonna come up with eight pie times five And then we just established that d a r d t was equal to one. So this is 40 high surface area is gonna increase at this area feet squared her minute. So that is the increase in the surface area of this balloon under these conditions that we have. So again, we started out with volume of a sphere, figured out how the radius was changing. And then once we knew how the radius was changing, we could see how the surface area was changing in this problem.

This is about a spherical balloon and is talking about volume. So volume from a balloon spiritualism before three. Q. Because that's the volume of sear. It tells us rate with changing. The house is the rate it's 100 boy, and then it's cubic feet per minute. All right? And it also says that the radius at the time were talking in my feet and want to know how radio increasing the arm by DT for part a What we're gonna do is take the derivative d v by d t He's gonna equal 4/3 by eyes and then three r squared and then times the are by ET okay, and again we're looking for this d r by d t. All right, But we're gonna plug in our d v by d t That's 100 pie. These threes canceled. So I just get four times pi times ours five Okay, squared and then times do you everybody t. So what that's going to give us is that 100 by is equal to 105. His four times five squared is four times 25 before. That's what if I was 100 pie de iron by 80. Which means if I divide these, if I can do all right, if I divide those, I'm going to end up with one. So the are mighty tea is one and one. What? Her minute. Okay, so we're actually gonna write this like this one foot per minute. So that's the Arab I t t now the surface area for part B is four pi r squared. So if we take the derivative of that, we get the essay by T t. That's what asking its boat. It's four pi time T o r times the our bodies. Okay, so that's gonna be eight pi times five and then times one. And that gives you 40 pie. I'm not sure I'm gonna have Room is barely. It's 40 pie day and I'll just right up here. So you get a little better in 40 pie, and then for a minute

Okay, so we've got a spherical balloon and it is being inflated. So volume is changing. DVD ci concentrate 100. That's in cubic feet. They're 100 pie, cute feet. Her minute. And the equation that relates the volume lovable to its radius is volume equals 4/3. Hi are huge. So if we wantto think about how the radius is changing, then we're gonna do the derivative of this with respect to tea. So Devi be thine equals 4/3. Why? Times three are square d r times d r d t Well, now we can plug in the 100 pie and salt for d a r d t So 100 pie equals three of the thirds will cancel on line up with for high times are square and were told to find when our is five so well times that by why square times d already t Well, we find out pretty quick that the rd G that's the old one. And so that's gonna be here, I think problem in it. Now, to find a surface area, we need to get a equation involving the surface area. So the surface area of a sphere is given Bernie equation for hi are square. That means that d a d t would be cool 45 times to our times d r the scene. And so if we wanted to know how it was changing at them when the radius was five feet Well, then do you need thio Equals four times pi times two times five times what we found d rt t to be English would be one. So that's gonna come out to be for you high and that's gonna be a square feet.

Based on the giving ways I switch. The spherical balloon is inflated with helium. You have T. V. Over D. C. To be equal 200 bye beads cube me. So we record that we we know we is equal to four divided by three. I are cute. So what we do is we differentiates this so we have T. V. Uber dixie C. B. Equals. So you have four divided by 35. So are we give us three R. Squared. They are over D. C. So you simplify and you have D. V. D. C. This will give us four by out squared the R. Over the sea. So these three and that cancels out now divide both sides by this. So you have they are you ever did see will be equal to D. V. Divided by D. C. Divided by four pi out squeak. So now we can substitute the V. To be equal to this. So we know Devi it's equal 200. Hi so you have funded by divided by four Pi outs quake. So you have this and that's castle south. And Our it's 5 to this will give us 100 100 divided by four. How It's five sq. And this is equal to one feeds. Uh huh. Me that is the our D. C. So we we record as the question that gives the surface area of his fear. So you notice that the surface area A. It's equal to four pi. I was squared Saudi air over the sea will be equal to eight by are they are over? Did see and we know what the R. A. S. We know what our is as well. So then this implies that you have is by our is five, Then the R. D. C. It's one. So this would be called to 14 Mm. This time this is 40 hey speed squared a meme. So then we conclude that's by saying that the surface area increases. It's this reads.


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