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Activity 11.3.4.

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{align*} C \amp= 1449.2+4.6T-0.055T^2+0.00029T^3 \\ \amp \quad +(1.34-0.01T)(S-35)+0.016D \text{.} \end{align*}

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.

(a)

State the units for each of the partial derivatives, \(C_T\text{,}\) \(C_S\) and \(C_D\text{.}\) For each partial derivative, write a sentence that explains the physical meaning of the partial derivatives in terms of the rate of change for \(C\text{.}\)

(b)

Use algebraic rules to find the partial derivatives \(C_T\text{,}\) \(C_S\) and \(C_D\text{.}\)

(c)

Evaluate each of the three partial derivatives at the point where \(T=10\text{,}\) \(S=35\) and \(D=100\text{.}\) 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 \((10,35, 100)\text{.}\)