Answer:
The percentage rate of decay per year is of 3.25%.
The function showing the mass of the sample remaining after t is [tex]A(t) = 80(0.9675)^t[/tex]
Step-by-step explanation:
Equation for decay of substance:
The equation that models the amount of a decaying substance after t years 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 21 years, its mass decreases by half.
This means that [tex]A(21) = 0.5A(0)[/tex]. We use this to find r, the percentage rate of decay per year.
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]0.5A(0) = A(0)(1-r)^{21}[/tex]
[tex](1-r)^{21} = 0.5[/tex]
[tex]\sqrt[21]{(1-r)^{21}} = \sqrt[21]{0.5}[/tex]
[tex]1 - r = 0.5^{\frac{1}{21}}[/tex]
[tex]1 - r = 0.9675[/tex]
[tex]r = 1 - 0.9675 = 0.0325[/tex]
The percentage rate of decay per year is of 3.25%.
Given that the initial mass of a sample of Element X is 80 grams.
This means that [tex]A(0) = 80[/tex]
The equation is:
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]A(t) = 80(1-0.0325)^t[/tex]
[tex]A(t) = 80(0.9675)^t[/tex]
The function showing the mass of the sample remaining after t is [tex]A(t) = 80(0.9675)^t[/tex]
