Answer:
The half-life of this radioactive substance is of 8.6 months.
Step-by-step explanation:
Exponential decay of an amount:
The equation that models an amount after t units of time, subject to exponential decay, is given by:
[tex]A(t) = A(0)(1-r)^t[/tex]
In which A(0) is the initial amount and r is the decay rate, as a decimal.
A radioactive substance decays to 30% of its original mass in 15 months.
This means that [tex]A(15) = 0.3A(0)[/tex]. We use this to find 1 - r.
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]0.3A(0) = A(0)(1-r)^{15}[/tex]
[tex](1-r)^{15} = 0.3[/tex]
[tex]\sqrt[15]{(1-r)^{15}} = \sqrt[15]{0.3}[/tex]
[tex]1 - r = (0.3)^{\frac{1}{15}}[/tex]
[tex]1 - r = 0.9229[/tex]
So
[tex]A(t) = A(0)(0.9229)^t[/tex]
Determine the half-life of this radioactive substance to the nearest tenth.
This is t for which A(t) = 0.5A(0). So
[tex]A(t) = A(0)(0.9229)^t[/tex]
[tex]0.5A(0) = A(0)(0.9229)^t[/tex]
[tex](0.9229)^t = 0.5[/tex]
[tex]\log{(0.9229)^t} = \log{0.5}[/tex]
[tex]t\log{0.9229} = \log{0.5}[/tex]
[tex]t = \frac{\log{0.5}}{\log{0.9229}}[/tex]
[tex]t = 8.6[/tex]
The half-life of this radioactive substance is of 8.6 months.
