1. Quote:
The speed of sound \displaystyle \begin{align*} c \end{align*} in an ideal fluid is related to the temperature \displaystyle \begin{align*} T \end{align*} (measured in \displaystyle \begin{align*} ^{\circ} K \end{align*}) by

\displaystyle \begin{align*} c = \sqrt{\gamma \, R \, T} \end{align*}

where \displaystyle \begin{align*} \gamma \end{align*} and \displaystyle \begin{align*} R \end{align*} are constants.

Suppose that \displaystyle \begin{align*} T \end{align*} increases by 10% from some base value. Use calculus to determine the approximate percentage change in \displaystyle \begin{align*} c \end{align*}, according to the ideal fluid model.
If we remember that the derivative is defined by \displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{\Delta\,x \to 0} \frac{y\left( x + \Delta\,x \right) - y\left( x \right)}{\Delta \, x} \end{align*} then that means that as long as \displaystyle \begin{align*} \Delta \,x \end{align*} is small, then \displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} \approx \frac{\Delta \,y}{\Delta\,x} \end{align*}.

Thus we can say \displaystyle \begin{align*} \frac{\mathrm{d}c}{\mathrm{d}T} \approx \frac{\Delta\,c}{\Delta\,T} \end{align*} and so \displaystyle \begin{align*} \Delta\,c \approx \frac{\mathrm{d}c}{\mathrm{d}T}\,\Delta\,T \end{align*}.

Now from \displaystyle \begin{align*} c = \sqrt{\gamma\,R\,T} \end{align*} we have \displaystyle \begin{align*} \frac{\mathrm{d}c}{\mathrm{d}T} = \frac{\sqrt{\gamma\,R}}{2\,\sqrt{T}} \end{align*}, and so with a 10\% increase in T, that means \displaystyle \begin{align*} \Delta\,T = \frac{T}{10} \end{align*}, thus

\displaystyle \begin{align*} \Delta\,c &\approx \frac{\sqrt{\gamma\,R}}{2\,\sqrt{T}}\cdot \frac{T}{10} \\ &= \frac{\sqrt{\gamma\,R\,T}}{20} \\ &= \frac{c}{20} \end{align*}

So that means that the change in c is \displaystyle \begin{align*} \frac{1}{20} \end{align*} of the original c, so a 5% increase.

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