Answer:
The percentage rate of decay per year is of 5.19%.
The function showing the mass of the sample remaining after t is [tex]A(t) = 520(0.9481)^t[/tex]
Step-by-step explanation:
Exponential equation of decay:
The exponential equation for the amount of a substance 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.
Every 13 years, its mass decreases by half.
This means that [tex]A(13) = 0.5A(0)[/tex]. We use this to find r. So
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]0.5A(0) = A(0)(1-r)^{13}[/tex]
[tex](1-r)^{13} = 0.5[/tex]
[tex]\sqrt[13]{(1-r)^{13}} = \sqrt[13]{0.5}[/tex]
[tex]1 - r = (0.5)^\frac{1}{13}[/tex]
[tex]1 - r = 0.9481[/tex]
[tex]r = 1 - 0.9481[/tex]
[tex]r = 0.0519[/tex]
The percentage rate of decay per year is of 5.19%.
The initial mass of a sample of Element X is 520 grams
This means that [tex]A(0) = 520[/tex]. So
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]A(t) = 520(1-0.0519)^t[/tex]
[tex]A(t) = 520(0.9481)^t[/tex]
The function showing the mass of the sample remaining after t is [tex]A(t) = 520(0.9481)^t[/tex]
