The temperature at a point (x, y, z) is given by T(x, y, z) = 300e-x² - 3y² - 72² where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature (in °C/m) at the point P(4, -1, 2) in the direction toward the point (5, -5, 5). °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 - 3y² - 72² where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature (in °C/m) at the point P(4, -1, 2) in the direction toward the point (5, -5, 5). °C/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
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The temperature at a point (x, y, z)is given by T(x, y, z) = 300e−x2 − 3y2 − 7z2 where T is measured in °C and x, y, z in meters.
Find the rate of change of temperature (in °C/m) at the point P(4, −1, 2)in the direction toward the point (5, −5, 5)
In which direction does the temperature increase fastest at P?
Find the maximum rate of increase at P.
![The temperature at a point \((x, y, z)\) is given by
\[ T(x, y, z) = 300e^{-x^2 - 3y^2 - 7z^2} \]
where \( T \) is measured in °C and \( x, y, z \) in meters.
(a) Find the rate of change of temperature (in °C/m) at the point \( P(4, -1, 2) \) in the direction toward the point \( (5, -5, 5) \).
\[ \boxed{\ \ \ } \ °\text{C/m} \]
(b) In which direction does the temperature increase fastest at \( P \)?
\[ \boxed{\ \ \ } \]
(c) Find the maximum rate of increase at \( P \).
\[ \boxed{\ \ \ } \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F225ea937-ec91-4f60-983d-e6a97001b794%2Fe13c1b37-b71a-4c9c-ba8b-66c5d4f98bf7%2Fwdn3hd_processed.png&w=3840&q=75)
Transcribed Image Text:The temperature at a point \((x, y, z)\) is given by
\[ T(x, y, z) = 300e^{-x^2 - 3y^2 - 7z^2} \]
where \( T \) is measured in °C and \( x, y, z \) in meters.
(a) Find the rate of change of temperature (in °C/m) at the point \( P(4, -1, 2) \) in the direction toward the point \( (5, -5, 5) \).
\[ \boxed{\ \ \ } \ °\text{C/m} \]
(b) In which direction does the temperature increase fastest at \( P \)?
\[ \boxed{\ \ \ } \]
(c) Find the maximum rate of increase at \( P \).
\[ \boxed{\ \ \ } \]
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