Chapter 5.6, Problem 2E

Speed of Sound in the North Atlantic The speed of sound in ocean water is $1148.94\text{meters}\text{per}\text{second}$, provided that the ocean water has a salinity of 35 $35\text{parts}\text{per}\text{thousand}$, the temperature it $0\text{degrees}\text{Celsius}$, and the measurement is taken at the surface. If any one of these three factors varies, the speed of sound also changes. Different oceans often differ in salinity. In the North Atlantic Central Water (the main body of water for the northern half of the Atlantic Ocean), the salinity can be determined from the temperature, so the speed of sound depends only on temperature and depth. A simplified polynomial formula for velocity in this body of water is

$\begin{array}{c}V=1447.733+4.7713T-0.05435T^2\\ +0.0002374T^3+0.0163D+1.675\times {10}^{-7}{D}^2\\ -7.139\times {10}^{-13}T{D}^3\end{array}$

Here $V$ is the speed of sound in meters per second, $T$ is the water temperature in degrees Celsius, and $D$ is depth in meters. This formula is valid for depths up to 8000 meters and for temperatures between $0$ and $30$ degrees Celsius.

a. What type of polynomial is $V$ as a function of $T$ alone? Of $D$ alone?

b. For a fixed depth of $1000\text{ meters}$, write the formula for $V$ in terms of $T$ alone.

c. Graph $V$ as a function of T for the fixed depth of $1000\text{ meters}$.

d. What is the concavity of the graph from part $c$? What does this imply about the speed of sound at that depth as temperature increases?