Suppose temperature °F in a region of 3-space, measured in meters, is given by T= 100 1+ (x − 3)² + y² + (z + 2)² and a bug is a point (2, 1,-1). (a) What direction should the bug fly when attracted to heat? (b) Find the components of a vector that starts at (2,1,-1) and ends at (3, 0, -2). Is this in the same direction as the převious answer? Physically, why fly towards (3, 0, -2)? (c) What is the rate of change of temperature at (2, 1, -1) when flying towards the heat? (decimal answer and include units) (d) What direction is the most rapid decrease in temperature?

Suppose temperature °F in a region of 3-space, measured in meters, is given by T= 100 1+ (x − 3)² + y² + (z + 2)² and a bug is a point (2, 1,-1). (a) What direction should the bug fly when attracted to heat? (b) Find the components of a vector that starts at (2,1,-1) and ends at (3, 0, -2). Is this in the same direction as the převious answer? Physically, why fly towards (3, 0, -2)? (c) What is the rate of change of temperature at (2, 1, -1) when flying towards the heat? (decimal answer and include units) (d) What direction is the most rapid decrease in temperature?

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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Question

3. Suppose temperature °F in a region of 3-space, measured in meters, is
given by
T =
100
1+ (x − 3)² + y² + (z + 2)²
and a bug is a point (2, 1,-1). (a) What direction should the bug fly
when attracted to heat? (b) Find the components of a vector that starts
at (2, 1, -1) and ends at (3, 0, -2). Is this in the same direction as the
previous answer? Physically, why fly towards (3,0,-2)? (c) What is the
rate of change of temperature at (2, 1, -1) when flying towards the heat?
(decimal answer and include units) (d) What direction is the most rapid
decrease in temperature?

Expert Solution

Step 1

Given:

$T = \frac{100}{1 + {(x + 3)}^{2} + y^{2} + {(z + 2)}^{2}}$

Where T is the temperature function.

Position of the bug = (2, 1, -1)

To find:

We have to find the direction of the bug when flying towards the heat.

We can find the direction of attraction to heat by finding the gradient of the temperature function by the following formula:

$\begin{array}{rcl} \nabla T & = & \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \end{array}$

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