We are looking for , and we know:
hence:
If we assume then and so:
We are given:
Can you put this together to finish?
Helium is pumped into a spherical balloon at . How fast is the radius increasing after 2 minutes?
I've searched some of the other related rates probs posted on this forum, so I apologize if a question like this has been brought up already.
Basically, I understand I'm looking for , and that .
Now, the rate is 2 and the volume of a sphere is .
Implicitly differentiating V with respect to T gets
I know that the rate the volume increases with respect to time is , and I'm not sure what to do with my 2 minutes (or 120 seconds). Is the volume with respect to time 120 seconds and is the rate with respect to time? That would make more sense I think because the dimensions of cubic feet with a factor of 2 is like a rate and seconds is most definitely a part of time.
However, when I try to solve or and that's incorrect. Since I don't know the volume with respect to time, how can I properly go about solving for the rate with respect to time?
I get
Do you mind if I ask where you derived that?
Well, I took the steps you laid out. since the problem asks for the rate of change with respect to time when t = 120 seconds, I plugged 120 into r(t) which == 3.86. Since we know the rate of change with respect to the volume we can plug that into dv/dr which gets 4pi(3.86)^2. And dV/dt is 2 cubic feet per second.
So the final equation came out to be which gave 0.01071 , and that means the growth of the balloon's volume has decreased significantly by the 2 minute mark. I'm just still unclear how you got the r(t) equation.
Thanks much.