# Maintaining constant ratio of diameters

We are inflating (with rate r1 & r2) two balloons by filling certain gasses as shown in figure.

The maximum diameter that the following balloons can attain (due to some restriction) is shown by the line segments AB and PQ. C and R are the mid point of the line segments and the centers of the balloons.

What is the necessary & sufficient condition to maintain the ratio of the diameters of these two balloons constant throughout out the process of expansion (till it attains maximum diameter) i.e:

   Dia(Big ballon)/Dia(small balloon)= Constant


What if I want to make sure that only final ratio and initial ratio (the state shown in fig) are same?

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What is varying? – copper.hat Nov 28 '12 at 8:08
Amount of gas inside and diameter. – gpuguy Nov 28 '12 at 8:17

## 1 Answer

Let $v_b, v_s$ be the volumes of the big & small balloons respectively, and $d_b, d_s$ be the corresponding diameters.

We have the relationships $d_b = \sqrt[3]{\frac{6}{\pi} v_b}$, $d_s = \sqrt[3]{\frac{6}{\pi} v_s}$.

If you wish to have $d_b = C d_s$, where $C$ is some constant, then you need to relate the volumes by $v_b = C^3 v_s$.

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I do not have any control over volume or even diameter. But I do have control over rates r1 & r2. – gpuguy Nov 28 '12 at 8:39
Rates of what? If it is flow rate then differentiate the volume constraint to get $\dot{v_b} = C^3 \dot{v_s}$. – copper.hat Nov 28 '12 at 9:17