The temperature at a point (x, y, z) is given by T(x, y, z) = 100e-x2 - Sy2 - 722 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 (6, -5, 4). -49 -2100e °C/m (b) In which direction does the temperature increase fastest at P? xi + 5yj + 7zk) (c) Find the maximum rate of increase at P. -18 1.617.10

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e-x2 - Sy2 - 722 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 (6, -5, 4). -49 -2100e °C/m (b) In which direction does the temperature increase fastest at P? xi + 5yj + 7zk) (c) Find the maximum rate of increase at P. -18 1.617.10

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 a point (x, y, z) is given by the equation T(x, y, z) = 100e-x^2-5y^2-7z^2 where T is measured in C and (x, y, z) is in meters. Find the rate of change of temperature at the point P(4, -1, 3) in the direction towards Q(5, -3, 5).
The temperature at a point (x,y,z) is given by T(x,y,z)=200e^(−(x^2)−(y^2)/4−(z^2)/9,) where T is measured in degrees celsius and x,y, and z in meters. Find the rate of change of the temperature at the point (1, 1, -2) in the direction toward the point (2, -3, -5).
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.
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.
The velocity (in centimeters per second) of blood molecules flowing through a capillary of radius 0.008 cm is v = 6.4 × 10−8 − 0.001r2, where r is the distance from the molecule to the center of the capillary. Find the rate of change of velocity with respect to r when r = 0.004 cm.
Suppose you are climbing a hill whose shape is given by the equationz= 1000−0.01x2−0.02y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (50, 80, 847). The positive x-axis points east and the positive y-axis points north. (a) If you walk northwest, will you start to ascend or descend? At what rate? (b) In what direction is the slope largest? Why?
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.
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?
Find the magnitude of the greatest rate of change of f(x, y, z) = (x2 + z2)3 at (1, 3, -2), and interpret geometrically.
Two straight roads intersect at right angles. A bicycle is moving towards the intersection along oneroad at a constant speed of 10 miles per hour. A man is walking away from the intersection alongthe other road at a constant speed of 5 miles per hour.(a) At what rate is the distance between the man and the bicycle changing when thebicycle is 3 miles away and the man is 4 miles away from the intersection?(b) Is the distance between the man and the bicycle increasing or decreasing at that instant
A car is behind a tractor on a single lane straight road with one lane in each direction. Both are moving at 15 m/s. The speed limit is 25 m/s, so the car wants to overtake. The safe distance between the car and the tractor is 20m. a) to overtake, the car goes onto the other side of the road and accelerates at a constant rate of 2 m/s^2 until reaching the speed limit, when it continues at constant speed. Show that the distance the car is ahead of the tractor at time t s after it starts to accelerate is given by t^2-20 for 0