The temperature at a point (x, y, z) is given by 400ex-3y2- 9z2 --_ (x, y, z) where T is measured in °C and x, y, z in meters (a) Find the rate of change of temperature at the point P(4, -1, 2) in the direction towards the point (5, -5, 6) oc/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

The temperature at a point (x, y, z) is given by 400ex-3y2- 9z2 --_ (x, y, z) where T is measured in °C and x, y, z in meters (a) Find the rate of change of temperature at the point P(4, -1, 2) in the direction towards the point (5, -5, 6) oc/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

The temperature at any point (x, y) on a steel plate is T = 500 − 0.6x2 − 1.5y2 , where x and y are measured in meters. At the point (2, 3), find the rates of change of the temperature with respect to the distances moved along the plate in the directions of the x- and y-axes.
The velocity (in centimeters per second) of blood molecules flowing through a capillary of radius 0.008 cmis v = 6.4 × 10−8 − 0.001r2, where r is the distance from the molecule to the center of the capillary. Findthe rate of change of velocity with respect to r when r = 0.004 cm.
A particle moves on the line y = 2x + 1 in such a way that its x−coordinate changes at a constant rate of 4 units per second. A right triangle is formed by the vertical line from the particle to the x−axis, the line connecting the particle to the origin, and the x−axis. At what rate is the area of the right triangle changing when x = 4?
The temperature at a point (x,y) on a flat metal plate is given by T(x,y)=60/(1+x2 +y2 ), where T is measured in °C and x,y in meters.Find the rate of change of temperature with respect to distance at the point (2,1) in (a) the x-direction and (b) the y-direction.
A rectangular shaped region is changing with time and increasing in the x direction at the rate of 2 units/sec and decreasing in the y direction at the rate of 0.5 units/sec. How does the diagonal change when x=4 and y=6? Is the diagonal increasing or decreasing at this moment?
Water is flowing at the rateof 6 m3/min from a reservoir shaped like a hemispherical bowl ofradius 13 m, shown here in profile. Answer the following questions,given that the volume of water in a hemispherical bowl ofradius R is V = (π/3)y2(3R - y) when the water is y meters deep. At what rate is the water level changing when the water is 8 m deep?
A small plane flies horizontally on a line 400 m directly above an observer with a speed of 70 m/s. Let θ be the angle of elevation of the plane. What is the rate of change of the angle of elevation, dθ/dt , when the plane is 500 m past the observer?
A runner starts on the east side of a circular track of radius 300 feet. Let’s say their inital (x, y)-coordinates are (300, 0). They run counterclockwise around the circle at a rate of 10π feet per second. What is the approximate rate of change of their x-coordinate at the 50-second mark?
Water is flowing at the rateof 6 m3/min from a reservoir shaped like a hemispherical bowl ofradius 13 m, shown here in profile. Answer the following questions,given that the volume of water in a hemispherical bowl ofradius R is V = (π/3)y2(3R - y) when the water is y meters deep. At what rate is the radius r changing when the water is 8 m deep?
A ladder 5 meters long is leaning against a vertical wall. Due to the smooth floor, the foot of the ladder is sliding away from the wall. Let x be the horizontal distance of the foot of the ladder from the wall and y be the height of the top of the ladder from the floor. Determine when will the rate of change of y with respect to x be numerically equal to the negative of the current value of x?
A rocket is blasting off from a launch pad and you are going to film the take off. Youposition yourself at a minimum safe distance of 4000 feet away. As the rocket launches straight up at1000 feet per second, you rotate the camera to keep the rocket in frame. Three seconds after thelaunch, determine both the rate at which the distance between you and the rocket is increasing, as wellas the rate at which the angle your camera’s line of sight creates with the horizontal is increasing. Ifneeded, round both measurements to the nearest 0.001 and include units.
A point is moving along the circle x2 + (y − 2)2 = 26 in such a way that its x-coordinate is decreasing at a rate of 3 unit per second. What is happening to the y-coordinate at the instant when the point has reached (−5, 3)? The y-coordinate is decreasing at a rate of ________ units per second.