Activity 11.4.4

The speed of sound $C$ traveling through ocean water is a function of temperature, salinity and depth. It may be modeled by the function $$\begin{aligned} C &= 1449.2+4.6T-0.055T^2+0.00029T^3 \\ & \quad +(1.34-0.01T)(S-35)+0.016D \end{aligned}$$. Here $C$ is the speed of sound in meters/second, $T$ is the temperature in degrees Celsius, $S$ is the salinity in grams/liter of water, and $D$ is the depth below the ocean surface in meters.

1. State the units for each of the partial derivatives, $C_T$, $C_S$ and $C_D$. For each partial derivative, write a sentence that explains the physical meaning of the partial derivatives in terms of the rate of change for $C$.

2. Use algebraic rules to find the partial derivatives $C_T$, $C_S$ and $C_D$.

3. Evaluate each of the three partial derivatives at the point where $T=10$, $S=35$ and $D=100$. Write a few sentences to explain what the sign of each partial derivatives tell us about the behavior of the function $C$ at the point $(10,35, 100)$.