Ant on a metal plate. The temperature at a point ( x, y) on a metal...

Ant on a metal plate. The temperature at a point ( x, y) on a metal plate is T(x, y) = 4x² - 4xy + y² . An ant on the plate walks around the circle of radius 10 centered at the origin.

a) What are the highest and lowest temperatures encountered by the ant?

b) Suppose the ant has changed its trajectory and is walking around the circle of radius 5. Is the highest temperature encountered by the ant greater or less compared to the one in part a)?

Expert Solution

milcah answered 2 years ago

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e−x2...

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e−x2 − 3y2 − 7z2 where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P(2, −1, 2) in the direction towards the point (4, −4, 4). answer in °C/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 T(x, y, z) = 300e−x2...

The temperature at a point (x, y, z) is given by T(x, y, z) = 300e−x2 − 3y2 − 9z2 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, 3) in the direction towards the point (6, −2, 6) (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

. [4 pts] The temperature of a 4 foot metal rod at a point x feet...

. [4 pts] The temperature of a 4 foot metal rod at a point x feet from its left endpoint is given by T(x) = 1 + 3x 2 , 0 ≤ x ≤ 4, where T is measured in degrees Celsius (and x is measured in feet). Find the average temperature of the rod. 5. [6 pts] A particle moves along a coordinate line and has an acceleration given by a(t) = 4t+ 6 cm/sec2 . If the particle’s...

The temperature T at a point (x,y,z) in space is inversely proportional to the square of...

The temperature T at a point (x,y,z) in space is inversely proportional to the square of the distance from (x,y,z) to the origin. It is known that T(0,0,1) = 500. a. [2] Compute T(2,0,0). b. [3] Find the rate of change of T at the point (2,3,3) in the direction of the point (3,1,1). c. [3] What is the maximal rate of change of T at the point (2,3,3)?

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...

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. just try to keep track of what needs to be a unit vector. a) Find the rate of change of the temperature at the point (1, 1, -1) in the direction toward the point (-5, -4, -3). b) In which direction (unit vector) does the temperature increase the fastest at (1, 1, -1)? c) What is...

(1 point) The temperature at a point (x,y,z) is given by ?(?,?,?)=200?−?2−?2/4−?2/9, where ? is measured...

(1 point) The temperature at a point (x,y,z) is given by ?(?,?,?)=200?−?2−?2/4−?2/9, where ? is measured in degrees Celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector. Find the rate of change of the temperature at the point (1, 1, 1) in the direction toward the point (-1, -1, -1). In which direction (unit vector) does the...

x' = x − y y' = 4y − x^2*y Linearize the system about the point...

x' = x − y y' = 4y − x^2*y Linearize the system about the point (2, 2). Classify the type and stability of the critical point at (2, 2) by examining the linearized system.

Find the steady state temperature function on a pi x pi rectangular plate, where the temperature...

Find the steady state temperature function on a pi x pi rectangular plate, where the temperature at the top edge is kept at sin(4x)oC and the remaining 3 sides are kept at 0oC

Find the point on the curve y=4x+4 closest to the point (0,8) (x,y)=

Find the temperature distribution at equilibrium in a rectangular plate (0 ≤ x ≤ L, 0...

Find the temperature distribution at equilibrium in a rectangular plate (0 ≤ x ≤ L, 0 ≤ y ≤ H) when the side at y = 0 is subject to the prescribed temperature g(x) = 2x, the sides at x = 0 and x = L are maintained at zero temperature, and the side at y = H is insulated, by using the method of separation of variables.

Question