Chapter 14, Problem 18RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

The speed of sound traveling through ocean water is a function of temperature, salinity, and pressure. It has been modeled by the function

*C* = 1449.2 + 4.6*T* − 0.055*T*^{2} + 0.00029*T*^{3} + (1.34 − 0.01*T*)(*S* − 35) + 0.016*D*

where *C* is the speed of sound (in meters per second), *T* is the temperature (in degrees Celsius), *S* is the salinity (the concentration of salts in parts per thousand, which means the number of grams of dissolved solids per 1000 g of water), and *D* is the depth below the ocean surface (in meters). Compute ∂*C*/∂*T*, ∂*C*/∂*S*, and ∂*C*/∂*D* when T = 10°C, *S* = 35 parts per thousand, and *D* = 100 m. Explain the physical significance of these partial derivatives.

To determine

**To compute:** The values of

Explanation

**Given:**

The speed of sound traveling through ocean water is,
*C* is the speed of sound (in meters per second), *S* is the salinity (the number of grams of dissolved solids per1000*g* of water) and *D* is the depth below the ocean surface (in meters).

**Calculation:**

The given function is,

Take the partial derivative with respect to *T* and obtain

Thus,

Substitute

Thus, the value of
*S* and *D* are constants.

Take the partial derivative with respect to *S* and obtain

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