The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug

Question

The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x = squareroot 1 + t, y = 2 + 1/3 t, where x and y are measured in centimeters. The temperature function satisfies T_x (2, 3) = 4 and T_y (2, 3) = 3. How fast is the temperature rising on the bug's path after 3 seconds? The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation C = 1449.2 + 4.6T - 0.055T^2 + 0.00029T^3 + 0.016D 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 a) Find dC/dt at t = 20 b) Interpret your answer to part a

Explanation / Answer

Solution:

dT/dt = (T/x) dx/dt + (T/y) dy/dt, by the Chain Rule
..............= (T/x) * (1/2)(1+t)^(-1/2) + (T/y) * (1/3).

At (x, y) = (2, 3) (and t = 3), we obtain {=> x = sqrt(1+t) => 2 = sqrt(1 + t) => t = 3}
dT/dt = 4 * (1/2)(1+3)^(-1/2) + 3 * (1/3) = 2 °C/sec.

After 3 seconds, temperature is rising at a rate of 2°C/s

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