A cup of water with a temperature of 95 ° C is placed in a room with a constant temperature of 21 ° C . (a) Assuming that Newton's Law of Cooling applies, use the result of Exercise 58 to find the temperature of the water t minutes after it is placed in the room. [ Note : The solution will involve a constant of proportionality.] (b) How many minutes will it take for the water to reach a temperature of 51 ° C if it cools to 85 ° C in 1 minute?
A cup of water with a temperature of 95 ° C is placed in a room with a constant temperature of 21 ° C . (a) Assuming that Newton's Law of Cooling applies, use the result of Exercise 58 to find the temperature of the water t minutes after it is placed in the room. [ Note : The solution will involve a constant of proportionality.] (b) How many minutes will it take for the water to reach a temperature of 51 ° C if it cools to 85 ° C in 1 minute?
A cup of water with a temperature of
95
°
C
is placed in a room with a constant temperature of
21
°
C
.
(a) Assuming that Newton's Law of Cooling applies, use the result of Exercise 58 to find the temperature of the water t minutes after it is placed in the room. [Note: The solution will involve a constant of proportionality.]
(b) How many minutes will it take for the water to reach a temperature of
51
°
C
if it cools to
85
°
C
in 1 minute?
A factorization A = PDP 1 is not unique. For A=
7 2
-4 1
1
1
5 0
2
1
one factorization is P =
D=
and P-1
30
=
Use this information with D₁
=
to find a matrix P₁ such that
-
-1 -2
0 3
1
-
- 1
05
A-P,D,P
P1
(Type an integer or simplified fraction for each matrix element.)
Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
30 -1
-
1 0 -1
400
0
0 1
A=
3 4 3
0 1 3
040
3 1 3
0 0
4
1
0
0
003
-1 0 -1
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
A basis for the corresponding eigenspace is {
A. There is one distinct eigenvalue, λ =
B. In ascending order, the two distinct eigenvalues are λ₁
...
=
and 2
=
Bases for the corresponding eigenspaces are {
and ( ), respectively.
C. In ascending order, the three distinct eigenvalues are λ₁ =
=
12/2
=
and 3 = Bases for the corresponding eigenspaces are
{}, }, and {
respectively.
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