Answer:
2.6 billion years
Explanation:
There are essentially two ways of solving nuclear half-life problems. One way is by applying the half-life formula, which is
A
(
t
)
=
A
0
(
t
)
⋅
(
1
2
)
t
t
1
2
, where
A
(
t
)
- the quantity that remains and has not yet decayed after a time t;
A
0
(
t
)
- the initial quantity of the substance that will decay;
t
1
2
- the half-life of the decaying quantity;
In this case, the rock contains
1/4th
of the orignal amount of potassium-40, which means
A
(
t
)
will be equal to
A
0
(
t
)
4
. Plug this into the equation above and you'll get
A
0
(
t
)
4
=
A
0
(
t
)
⋅
(
1
2
)
t
t
1
2
, or
1
4
=
(
1
2
)
t
t
1
2
This means that
t
t
1
2
=
2
, since
1
4
=
(
1
2
)
2
.
Therefore,
t
=
2
⋅
t
1
2
=
2
⋅
1.3 = 2.6 billion years
A quicker way to solve this problem is by recognizing that the initial amount of the substance you have is halved with the passing of each half-life, or
t
1
2
.
This means that you'll get
A
=
A
0
2
after the first 1.3 billion years
A
=
A
0
4
after another 1.3 billion years, or
2
⋅
1.3 billion
A
=
A
0
8
after another 1.3 billion years, or
2
⋅
(
2
⋅
1.3 billion
)
