(7 pts) The temperature at a point (x,y,z) is given by T(x, y, z) = 200e^x^2-y^2
ID: 2865507 • Letter: #
Question
(7 pts) 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. 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, 2) in the direction toward the point (3, 1, -2). In which direction (unit vector) does the temperature increase the fastest at (-1, 1, 2)? What is the maximum rate of increase of T at (-1, 1, 2)?Explanation / Answer
T(x,y,z) = 200e^(-x^2 - y^2/4 - z^2/9)
(a)
?T = < -400x e^(-x^2 - y^2/4 - z^2/9) , -100y e^(-x^2 - y^2/4 - z^2/9) , -400z/9 * e^(-x^2 - y^2/4 - z^2/9) >
?T(-1, 1, 2) = < 400e^(-1 - 1/4 - 4/9) , -100 e^(-1 - 1/4 - 4/9) , -800/9 e^(-1 - 1/4 - 4/9) >
?T(-1, 1, 2) = < 400e^(-61/36) , -100 e^(-61/36) , -800/9 e^(-61/36) >
Vector from (-1, 1, 2) to (3, 1, -2) = <3 + 1, 1 - 1, -2 - 2 > = <4, 0, -4>
Unit vector in direction toward point (3, 1, -2)
u = <4, 0, -4> / || <4, 0, -4> ||
u = <4, 0, -4> / 4sqrt(2)
u = <1/sqrt(2), 0, -1/sqrt(2)>
Rate of change of temperature at point (-1, 1, 2) in direction toward point (3, 1, -2)
= ?T(-1, 1, 2) . u
= < 400e^(-61/36) , -100 e^(-61/36) , -800/9 e^(-61/36) > . <1/sqrt(2), 0, -1/sqrt(2)>
= [400/sqrt(2) + 0 + 800/9sqrt(2)]*e^(-61/36)
= 4400/9sqrt(2) * e^(-61/36)
= 63.515
(b)
Temperature increases fastest in direction of gradient at (-1, 1, 2)
= < 400e^(-61/36) , -100 e^(-61/36) , -800/9 e^(-61/36) >
= 100e^(-61/36) <4, -1, -8/9 >
Unit vector in direction of fastest increase
= <4, -1, -8/9 > / || <4, -1, -8/9 > ||
= <4, -1, -8/9 > / sqrt(1441/81)
= 9 *<4, -1, -8/9 > / sqrt(1441)
= <36, -9, -8> / sqrt(1441)
= <36 / sqrt(1441), -9 / sqrt(1441), -8 / sqrt(1441)>
(c)
Maximum rate of increase at P
= < 400e^(-61/36) , -100 e^(-61/36) , -800/9 e^(-61/36) > . <36 / sqrt(1441), -9 / sqrt(1441), -8 / sqrt(1441)>
= [14400 + 900 + 6400] * e^(-61/36) / sqrt(1441)
= 105.01
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