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Point) The radius of spherical balloon increasing at a rate of _ centimeters per minute_ How fast is the volume changing when the radius is 12 centimeters? Note: Th...

Question

Point) The radius of spherical balloon increasing at a rate of _ centimeters per minute_ How fast is the volume changing when the radius is 12 centimeters? Note: The volume of a sphere is given by V = (4/3)Ir"_Rate of change of volume

point) The radius of spherical balloon increasing at a rate of _ centimeters per minute_ How fast is the volume changing when the radius is 12 centimeters? Note: The volume of a sphere is given by V = (4/3)Ir"_ Rate of change of volume



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

As air is pumped into a spherical balloon, the radius is going to increase at a constant rate of two cm/s. So that's going to be the radius is increasing. So um two cm per second Is the change in radius with respect to time. TRT equals two cm/s. And we want to know the rate of change of the balloons volume when the rate is 10 m. So we want to know the rate of the changing volume with respect to the radius. So if you get here we need to find D V. D. T. Because if we can do D V D. T. Um then we'll end up being able to find um what we're looking for or actually need to find D V. D. R. That is so D V. D. R. We need to relate the volume and the radius though. And since it's a sphere we have the equals four thirds Hi are cube. And that's going to be our final answer.

Atmosphere V. Easy calls to for over three. Hi are cute. Right? So to find the rate of change, you have to find the derivative of this equation. Um With respect to time T. So D. V. Over D. T. So um the derivative of the volume with respect to time is equals two. Um So we fact out for over three pie because it's a constant and then we find the derivative of um Our cubes and that's going to be three R squared. Yeah they are over D. T. Yeah And this is equals two. Excuse me for pie? Our squid. They are over D. T. So it's clear that the two unknowns here is R. And D. Are over T. T. So to actually get that um we are given that the rate of exchange of the surface area with respect to time is actually equals to four. And we know that um the derivative of, Sorry we know that The formula for the surface area is equal to four pi our squid. So now we need to find the derivative with respect to time of um This equation so D. A. Divided by D. T. So the curative of the area with respect to time is equal to four pipe time. The director of R squared. So that's going to be times to our they are over D. Uh t. Yeah. Uh So we're also told that the rate of change of the radios with respect to time Is equals to 0.1. Okay so let's substitute that in. Let's first substitute um The deer is of of the area with respect to time so that's full equals two for pie times. Careless. Just simplify this so it's going to be eight pi are uh times the er over the th 0.1. So now we're in a position to actually so for our so our O. B. Equals two for over 85 Time 0.1. And this would be equals to five over. Hi. So now we have um the value of the arab Aditi and the value of our so we can come back to our equation here And make the substitution. So this would be equals two for pie. Um Rse calls to five over high squared The Arab ATTZ equals to 0.1. So um when you actually simplify all of this, you're going to get 10 over. Hi. So the derivative of the volume with respect to time physical to turn pie centimeters cube per minute. Mhm.

Hey there, this is a really cool conceptual question. So let's think about what we know about a sphere. Its volume is four thirds pira cute. So I'm going to make a table that has the time in this case, in seconds. Starting at time zero. The radius which is in inches. And we're told that that zero to start with and then the volume which will be an inch is cubed and that would be zero as well. Right? If the radius is zero. So our goal is to advance to three seconds. Let's see how to get there. Well after one second we know the rate of the radius is five inches per second. After one second it would be up to five. Well, what volume would we have for a sphere of radius? 0.25 or 1/4? That would be four thirds pi 1/4 cube. And if you work out the details, you'll see that. That's pi over 48. Okay, so let's advance another second while the radius changes by another 0.25 inches. So that would be 0.5 or one half. The volume. Men is four thirds pi one half cube. And that works out to 1/6 of pi five or six and then going up another second. The radius increases by another 0.25 So our radius now is 3/4. The volume would be four thirds pi 3/4 cube which works out to nine pi over 16. So let's see what the rate of change of volume is at time three seconds. Well the rate of change of volume would be how much the volume has changed across that one second. So that's an inches cube. And the time that's changed is from 2 to 3 seconds. So that's just for one second and if you do some fraction work 9/16 of pi minus 1/6 of pie is 19 pi over 48 cubic inches per second.


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