Question

The radius of a spherical balloon a is increasing at a constant rate of 0.1 cm/s. another balloon, b, is being continuously deflated in such a way that the total volume of a and b remains constant. suppose that at the beginning the radius of a is 10 cm and that of b is 9 cm. find the rate of change of the radius of b when the radius of a is 12 cm

Answer 1

Let
`r_A` and
`r_B` be the respective radii of balloons A and B. If the fixed total volume is V, then

`V = \frac{4\pi}3\left({r_A}^3 + {r_B}^3\right)`

and knowing
`r_A=10\,\rm cm` and
`r_B=9\,\rm cm` at the start, we have V = 6916π/3 cm³. Then when
`r_A=12\,\rm cm`, the radius of the other sphere is
`r_B=1\,\rm cm`.

Differentiating both sides with respect to time t gives a relation between the rates of change of the radii:

`0 = 4\pi \left({r_A}^2 (dr_A)/(dt) + {r_B}^2 (dr_B)/(dt)\right) \implies (dr_B)/(dt) = -\left((r_A)/(r_B)\right)^2 (dr_A)/(dt)`

We're given
`(dr_A)/(dt) = 0.1(\rm cm)/(\rm s)` the whole time. At the moment
`r_A=12\,\rm cm`, the radius of balloon B is changing at a rate of

`(dr_B)/(dt) = -\left((12\,\rm cm)/(1\,\rm cm)\right)^2 \left(0.1(\rm cm)/(\rm s)\right) = \boxed{-14.4 (\rm cm)/(\rm s)}`