Question

Linear Approximation problem for calculus

Answer 1

The volume of spherical balloon changes as air is pumped into the balloon and released from the balloon. The radius of the balloon, in cm, is modeled by a twice differential function r(t)=-t^2+4t-1, where t is measured in seconds, over the time interval from 0 to 5 seconds. Estimate the volume of the balloon at t=3.1 seconds using the tangent line approximation at t=3 seconds. Note that the volume of a sphere of radius r is given by V=(4/3)pi r^3

Answer 2

@bahrom7893

Answer 3

I found the dv/dt at t=3 is -32 pi

Answer 4

I'm having problems figuring out why the key is using (3, (32/2 pi) as their x and y value

Answer 5

maybe I am using the incorrect equation for linear approximation

Answer 6

What's the linear approximation formula again? sorry it's been a really long time since i last looked at this.

Answer 7

Let me look it up. I thought I could just use the tangent line
y-y1=m(x-x1)

Answer 8

@iPwnBunnies will enjoy solving this.

Answer 9

L(x)=f(a)+f ' (a)(x-a)

Answer 10

yup, that looks more like it.

Answer 11

Linear approximation is lame man.

Answer 12

But you love math

Answer 13

yes but I still need to do this problem. I would appreciate any help, really

Answer 14

L(x)=f(a)+f ' (a)(x-a) = f(3) + f'(3)(3.1-3)

Answer 15

they are using (3, (32/3)pi) but I don't know where (32/3) pi comes from

Answer 16

I was thinking about something like what Bahrom wrote. But I then realized it had to do with the volume of the sphere.

Answer 17

I think f' would be dv/dt at t = 3, but it's been over a year since i looked at cal

Answer 18

Ohhh, r(t) is the radius of the balloon. I thought it was the rate of the radius. :o

Answer 19

dv/dt evaluated at t=3 is -32pi I did understand that part
Am I using r(t) to find (3, ?)?

Answer 20

Maybe I need to integrate r(t) since it is twice differentiable

Answer 21

did you forget to multiply by dr/dt?

Answer 22

Yeah, it's strange. The wording for r(t) is not clear at all.

Answer 23

ok let me check that

Answer 24

Doesn't really say if it's the radius, the 2nd derivative of the radius or what.

Answer 25

V' = 4*pir^2 * dr/dt
dr/dt = r', etc.

Answer 26

no I did that, and I got -32pi

Answer 27

\[y-\frac{ 32\pi }{ 3 }=32\pi(t-3)\]

Answer 28

I am still not sure where the (32/3) pi comes from

Answer 29

this is what they are claiming is the next step. Then the last step uses t = 3.1 in the equation to say y is approximately 23.457

Answer 30

is that just some simplification?

Answer 31

from where? I don't see where the 3 was used. I thought that 3 was used in r(t) but I am not sure

Answer 32

wait I see it. They used r(3) to find 2 and then used 2 in the Volume formula

Answer 33

ok I got it, but I did not see it at all. Thanks for your help

Answer 34

Sorry, wasn't really helpful

Answer 35

that's ok, you have given me assistance before. Sometimes when you write it down and think about it, the solutions come anyways.