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)?

Expert Solution

Colby Messinger answered 1 year 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.

The temperature T at (x,y,z) in the 3D space is given by T(x,y,z) = ln(1+x2y2+z2). a)...

The temperature T at (x,y,z) in the 3D space is given by T(x,y,z) = ln(1+x2y2+z2). a) Find rate of change of T at the point P(1,-1,-1) in the direction of Q(2,0,0)? b) In which direction from P(1,-1,-1) does the temperature T increase most rapidly? c) What is the maximum rate of change of T at P(1,-1,-1)?

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...

The temperature T in a metal ball is inversely proportional to the distance from the center...

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120°. (a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (3, 3, 4).

The temperature T in a metal ball is inversely proportional to the distance from the center...

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120 degrees F . a) Find the rate of change of T at (1, 1,2) in the direction towards the point (2, 1,3). b) Show that at any point in the ball the direction of greatest increase in the temperature is given by a vector...

The temperature T in a metal ball is inversely proportional to the distance from the center...

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 160°. (a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (3, 1, 4). (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points...

The temperature function (in degrees Fahrenheit) in a three dimensional space is given by T(x, y,...

The temperature function (in degrees Fahrenheit) in a three dimensional space is given by T(x, y, z) = 3x + 6y - 6z + 1. A bee is constrained to live on a sphere of radius 3 centered at the origin. In other words, the bee cannot fly off of this sphere. What is the coldest temperature that the bee can experience on this sphere? Where does this occur? What is the hottest temperature that the bee can experience on...

Find the coordinates of the point (x, y, z) on the plane z = 4 x...

Find the coordinates of the point (x, y, z) on the plane z = 4 x + 1 y + 4 which is closest to the origin.

(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...

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

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Related Solutions

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) = 300e−x2...

The temperature T at (x,y,z) in the 3D space is given by T(x,y,z) = ln(1+x2y2+z2). a)...

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 T in a metal ball is inversely proportional to the distance from the center...

The temperature T in a metal ball is inversely proportional to the distance from the center...

The temperature T in a metal ball is inversely proportional to the distance from the center...

The temperature function (in degrees Fahrenheit) in a three dimensional space is given by T(x, y,...

Find the coordinates of the point (x, y, z) on the plane z = 4 x...

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