Solving Exponential Decay, Matrix Operations and Equations

5)
Answer :
1

A(t) = 400e−0.045t
At t = 0
A(0) = 400
Therefore, the original amount is 400. Half of it is 200. Therefore, at half
the original amount:
200 = 400e−0.045t
0.5 = e−0.045t
[Taking log on both sides]
ln 0.5 = −0.045t
−0.6931 = −0.045t
=⇒ t = 15.40
Thus, the substance decays to half its original amount in 15.4 years.
6)
x − 3y = 9
−2x − 2y = 6
Answer:
Matrix form: A · x = B
[ 1 −3
−2 −2
]
︸ ︷︷ ︸
A
[x
y
]
︸︷︷︸
x
=
[9
6
]
︸︷︷︸
B
The boxed element (1) is the pivot. Augmented matrix [A|B] is:
[A|B] =
[ 1 −3 | 9
−2 −2 | 6
]
[ Multiply Row 1 (R1) by 2 and add to Row 2 (R2) : R2 = R2 + 2 × R1]
=
[1 −3 | 9
0 −8 | 24
]
2