Math 241
Name: __________________________________
Homework 5
1. A copper rod with a length of )!.! cm is placed along the B-axis, with the left end at B œ !. The
middle portion of the rod is heated, and then the rod is thermally insulated from its surroundings.
(a) After being heated, the initial temperature distribution of the rod is given by the formula:
X aBb œ E  F cosaGBb
where E œ %#Þ! °C, F œ ##Þ! °C, and G œ !Þ!()& radiansÎcm. Sketch a graph of this
distribution on the following axes:
FINAL ANSWER
70
60
60
Temperature H°CL
Temperature H°CL
PRACTICE SPACE
70
50
40
30
20
0
20
40
60
Position HcmL
80
50
40
30
20
0
20
40
60
Position HcmL
(b) What is the initial temperature at the center of the rod? What is the initial temperature at the
ends of the rod?
80
According to the physics of heat conduction, the temperature X aBß >b of the rod should change over
time according to the following partial differential equation:
`X
` #X
œ ! #,
`>
`B
where ! œ '(Þ% cm# Îmin for copper. This PDE is known as the heat equation.
(c) According to the heat equation, how quickly will the temperature at the center point of the rod
initially decrease?
(d) Find the value of - for which the function
X aBß >b œ E  F/-> cosaGBb,
is a solution to the heat equation.
(e) Assuming X aBß >b has the form given in part (d), make a rough contour plot of the function
X aBß >b on the following axes. Include contours for 25 °C, 30 °C, 35 °C, 40 °C, 45 °C, 50 °C,
55 °C, and 60 °C. (You will probably need to use a graphing calculator or a computer for this.)
FINAL ANSWER
5
4
4
Time HminL
Time HminL
PRACTICE SPACE
5
3
2
1
3
2
1
0
0
0
20
40
60
Position HcmL
80
0
20
40
60
Position HcmL
80
(f) Sketch a graph of the temperature distribution of the rod at time > œ $Þ!! min.
FINAL ANSWER
70
70
60
60
Temperature H°CL
Temperature H°CL
PRACTICE SPACE
50
40
30
20
0
20
40
60
Position HcmL
80
50
40
30
20
0
(g) What is the equilibrium temperature of the rod as > Ä _?
20
40
60
Position HcmL
80
2. We are given the following information about a function 0 aBß Cb:
`0
œ $B#  #B/C ,
`B
`0
œ B# /C  #C,
`C
0 a"ß !b œ %.
Based on this information, find a formula for 0 aBß Cb in terms of B and C.
3. The following table shows some values for a function 0 aBß Cb:
B
C
%Þ'
%Þ)
&Þ!
&Þ#
&Þ%
$Þ%
#Þ'%
$Þ!'
$Þ%%
$Þ()
%Þ!)
$Þ#
#Þ"'
#Þ&#
#Þ)%
$Þ"#
$Þ$'
$Þ!
"Þ)%
#Þ"%
#Þ%!
#Þ'#
#Þ)!
#Þ)
"Þ')
"Þ*#
#Þ"#
#Þ#)
#Þ%!
#Þ'
"Þ')
"Þ)'
#Þ!!
#Þ"!
#Þ"'
(a) Use this data to estimate the value of
` #0
at the point a&ß $b.
`B#
(b) Use this data to estimate the value of
` #0
at the point a&ß $b.
`B `C