Answer
$v=\frac{1087}{\sqrt{273}} T^{\frac{1}{2}} f t / s$
Work Step by Step
\[
\frac{d v}{d T}=\frac{1087}{2 \sqrt{273}} T^{\frac{1}{2}} \Rightarrow d v=\frac{1087}{2 \sqrt{273}} T^{-\frac{1}{2}} d T
\]
We get:
\[
\begin{array}{c}
\int d v=\int \frac{1087}{2 \sqrt{273}} T^{-\frac{1}{2}} d T \\
v=\frac{1087}{\sqrt{273}} T^{\frac{1}{2}}+C
\end{array}
\]
The speed of sound in air at $0 \mathrm{C}$ (or $273 \mathrm{K}$ on the Kelvin scale) is $1087 \mathrm{ft} / \mathrm{s}$:
$\frac{1087}{\sqrt{273}} T^{\frac{1}{2}}+C=v\Rightarrow \frac{1087}{\sqrt{273}}(273)^{\frac{1}{2}}+C=1087$
$1087=1087+C$
$\therefore C=0$
$v=\frac{1087}{\sqrt{273}} T^{\frac{1}{2}} f t / s$
