   Chapter 14.5, Problem 37E

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Textbook Problem

# The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation C   = 1449.2   +   4.6 T −   0.055 T 2 +   0.00029 T 3 +   0.016 D where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver’s depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units? To determine

To estimate:

The rate of change (with respect to time) of the speed through the ocean water experienced by the diver 20 minutes into the dive and its units

Explanation

1) Concept:

By using chain rule, we find the rate of change of the speed through the ocean water

The Chain Rule:

Suppose that z=f(x, y) is a differentiable function of x and y, where x=g(t) and y=ht are both differentiable function of t and

dfdt= fxdxdt+fydydt

2) Given:

C=1449.2+4.6T-0.055T2+0.00029T3+0.016D

3) Calculation:

As the speed of sound traveling through the ocean water with salinity 35 parts per thousand has been modeled as

C=1449.2+4.6T-0.055T2+0.00029T3+0.016D

Differentiating C with respect to T,

CT=4.6-0.11T+0.000877T2

Similarly, Differentiating C with respect to D

CD=0.016

Therefore, the rate of change of the speed of sound through the ocean water experienced by the diver into the dive is obtained by differentiating C with respect to t.

Therefore, by applying chain rule,

dCdt= CTdTdt+CDdDdt

From given graph,

At t=20 minutes, temperature is T=12

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