The temperature will decrease at approximately 27.72 degrees per second as Captain Ralph moves in the direction specified at a speed of e^9 meters per second.
The rate of change of temperature as Captain Ralph moves in a specific direction is determined by the gradient of the temperature function. Given the temperature function T(x,y,z)=
, the gradient vector ∇T represents the direction of the steepest increase in temperature. The magnitude of ∇T at the point (1, 1, 1) indicates the rate of change of temperature in that direction.
When Captain Ralph travels in the direction specified at a speed of e^9 meters per second, the rate of temperature decrease is approximately 27.72 degrees per second, as determined by the magnitude of the gradient vector at his current location.
Complete ques:
Captain Ralph is in trouble near the sunny side of Mercury. The temperature of the ship's hull when he is at location (x, y, z) will be given by T (x, y, z) = e^(−x^2 − 2y^2 − 3z^2), where x, y, and z are measured in meters. He is currently at (1, 1, 1). If the ship travels at e9 meters per second, how fast (in degrees per second) will the temperature decrease if he proceeds in that direction?