Question

PROBLEM STATEMENT:

You are blowing air into a balloon at a rate of 4*pi/3 cubic inches per second. (The reason for this strange-looking rate is that it will simplify your algebra a little bit.)

Assume the radius of your balloon is zero at time zero.

Let r(t), A(t) and V(t) denote the radius, surface area and the volume of your balloon at time t, respectively. (Assume the thickness of the skin is zero.)

Find:

a) r'(t)

b) A'(t)

c) V'(t)

Answer 1

MY ATTEMPT OF THE PROBLEM:

I know that dV/dt = 4*pi/3 and that dV/dt = 4*pi * r^2 dr/dt, and that 4*pi/3 = 4*pi r^2 * dr/dt, which implies that 1/3 = r^2 * dr/dt.

I also found that dA/dt = 8*pi*r * dr/dt.

My issue is that I now have two equations, 1/3 = r^2 * dr/dt and dA/dt = 8*pi*r * dr/dt, but three unknowns, dr/dt, dA/dt and r.

I'm assuming that I need to find a third relationship/equation, but I cannot figure out what it is.

As always, any help would be very much appreciated!

Answer 2

I agree with part a)
dr/dt = 1 / (3r^2)

Answer 3

for part b, you can substitute

b) dA/dt = 8 Pi * r * dr/ dt = 8 Pi * r * 1 / (3r^2)

Answer 4

I think they want each quantity as a function of "t"

Answer 5

I remember having successfully done this problem a long time ago, and it was a system of equations, where I found r'(t), A'(t) as functions of t alone.

So, dA/dt = 8/3 * pi / r, but then how do I get a relationship with the radius and time?

Answer 6

dV/dt = 4pi/3
dV = 4pi/3 dt
integrate
V= 4pi/3 t + C
r is 0 at t=0 so that means C=0
V= 4pi/3 t
use that to find
4pi/3 t = 4pi/3 r^3
and solve for r in terms of t
then find dr/dt in terms of t

Answer 7

This is for a course that has yet to cover integrals, though.

Answer 8

we don't have to integrate (because the rate is constant)
in other words we can use rate (of increase in volume) * time = volume
thus V= 4 pi/3 * t = 4pi/3 r^3

Answer 9

Alright, so I think I get it!

Thanks! :D

Answer 10

dv/dt is constant (independent of time) = 4pi/3

r = t^(1/3)
dr/dt = (1/3) t^(-2/3)

use that in
dA/dt = 8 pi r dr/dt

Answer 11

perl has given the answers
dv/dt==4pi/r
dr/dt= 1/3r^2
da/dt =8pi/3r