A spherical balloon is being inflated with helium gas from empty at a constant rate of $100\pi$ \(

Question

A spherical balloon is being inflated with helium gas from empty at a constant rate of $100\pi$ $\text{cm}^3 / \text{s}$. Find the radius, $r$ $\text{cm}$, and the total surface area, $A$ $\text{cm}^2$, of the balloon after six minutes. Hence find, respectively, the rates of change of $r$ and $A$ at this instant. After the balloon is inflated, it is released into the atmosphere. Given that the volume of the balloon, $V$ $\text{m}^3$, is related to the pressure, $P$ $\text{atmosphere}$, by the equation $PV = 2.4$, find the rate of change of $V$ with respect to $P$ when $V = 2$ $\text{m}^3$.

Accepted Answer

(i) $r=30\text{ cm},\ A=3600\pi\text{ cm}^2$ (ii) $\dfrac{\mathrm{d}r}{\mathrm{d}t}=\dfrac1{36}\text{ cm s}^{-1},\ \dfrac{\mathrm{d}A}{\mathrm{d}t}=\dfrac{20\pi}{3}\text{ cm}^2\text{ s}^{-1}$ (iii) $\dfrac{\mathrm{d}V}{\mathrm{d}P}=-\dfrac53\text{ m}^3\text{ atmosphere}^{-1}$

Differentiation - Connected Rates of Change

Differentiation - Connected Rates of Change

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