Question

Estimate the rate of change of the speed of sound\(C = 1449.2 + 4.6T - 0.055{T^2} + 0.00029{T^3} + 0.016D\) experienced by the driver 20 minutes into the dive and define its unit.

Short answer

The speed of sound experienced by the driver decreasing approximately\(0.32(\;{\rm{m}}/{\rm{s}})/{\rm{min}}\).

Step-by-step solution

Chain rule

"Suppose that\(z = f(x,y)\)is a differentiable function of\(x\)and\(y\), where\(x = g(t)\)and\(y = h(t)\)are both differentiable functions of\(t\). Then,\(z\)is differentiable function of\(t\)and\(\frac{{dz}}{{dt}} = \frac{{dz}}{{dx}} \cdot \frac{{dx}}{{dt}} + \frac{{dz}}{{dy}} \cdot \frac{{dy}}{{dt}}\)”

Step 2: Estimate the rate of change of the speed of sound

As given, the speed of sound is, \(C = 1449.2 + 4.6T - 0.055{T^2} + 0.00029{T^3} + 0.016D\)

Where, \(T\) is temperature in degree Celsius. \(D\) is depth below the ocean surface in meters.

Apply chain rule case 1

\(\begin{aligned}{l}\frac{{dC}}{{dt}} = \frac{{\partial C}}{{\partial T}}\frac{{dT}}{{dt}} + \frac{{\partial C}}{{\partial D}}\frac{{dD}}{{dt}}\\\frac{{dC}}{{dt}} = \left( {4.6 - 0.11T + 0.00087{T^2}} \right) \cdot {T_t} + 0.016 \cdot {D_t}\end{aligned}\)

At the instant \(t = 20\),

\(\frac{{dC}}{{dt}}(20) = \left( {4.6 - 0.11 \cdot T(20) + 0.00087 \cdot T{{(20)}^2}} \right) \cdot {T_t}(20) + 0.016 \cdot {D_t}(20)\)

By using the given information on the graphics of the functions \(T(t)\) and \(D(t)\), estimate \({T_t}(20)\) and \({D_t}(20)\) as follows,

\(\begin{aligned}{l}\frac{{dT}}{{dt}}(20) \approx \frac{{T(25) - T(20)}}{{25 - 20}}\\\frac{{dT}}{{dt}}(20) = \frac{{12 - 12.5}}{5}\\\frac{{dT}}{{dt}}(20) = - 0.1,\\\frac{{dD}}{{dt}}(20) \approx \frac{{D(25) - D(20)}}{{24 - 20}}\\\frac{{dD}}{{dt}}(20) = \frac{{10 - 7}}{4}\\\frac{{dD}}{{dt}}(20) = 0.75.\end{aligned}\)

Calculate \({C_t}(20)\)

\(\begin{aligned}{l}\frac{{dC}}{{dt}}(20) = \left( {4.6 - 0.11 \cdot T(20) + 0.00087 \cdot T{{(20)}^2}} \right) \cdot {T_t}(20) + 0.016 \cdot {D_t}(20)\\\frac{{dC}}{{dt}}(20) = - 0.324\end{aligned}\)

Thus, the speed of sound experienced by the driver decreasing approximately\(0.324(\;{\rm{m}}/{\rm{s}})/{\rm{min}}\).