Answer:
f(x)=890(98.68)^t-1
Step-by-step explanation:
the amount decays by 1.32% each year. So multiply by 98.68%, which is the remaining amount after 1 year.
f(x)= 890(98.68)^t-1
890 is your initial amount. Multiply that by 98.68 to get the remaining amount after 1 year. then multiply that to the power of t-1, t being the term number, or amount of years from the initial amount. You subtract by 1 because 890(98.68) is already demonstrating the amount after the first term.
Function representing the decay of a radioactive element will be [tex]A(t)=890(1-\frac{1.81}{100})^t[/tex] and the percentage rate of decay will be 1.81%.
Decay of a radioactive element is given by the function,
[tex]A(t)=A_0(1-\frac{r}{100})^t[/tex]
Here, [tex]A(t)=[/tex] Final amount of the radioactive element
[tex]A_0=[/tex] Initial amount
[tex]r=[/tex] Rate of decay
[tex]t=[/tex] Duration of decay or time
If [tex]A_0=890[/tex] grams
Function representing the decay will be,
[tex]A(t)=890(1-\frac{r}{100})^t[/tex]
Since, Element 'X' decreases by half of the original amount after every 38 years,
Therefore, [tex]\frac{890}{2} =890(1-\frac{r}{100})^{38}[/tex]
[tex]\frac{1}{2}=(1-\frac{r}{100})^{38}[/tex]
By taking log on each side of the equation,
[tex]\text{log}(\frac{1}{2})=\text{log}(1-\frac{r}{100})^{38}[/tex]
[tex]-\text{log}(2)=\text{(38)log}(1-\frac{r}{100})[/tex]
[tex]-0.007922=\text{log}(1-\frac{r}{100})[/tex]
[tex]10^{-0.007922}=1-\frac{r}{100}[/tex]
[tex]0.98192=1-\frac{r}{100}[/tex]
[tex]r=100(0.018075)[/tex]
[tex]r\approx1.81[/tex]%
Therefore, function representing the decay of the radioactive element will be [tex]A(t)=890(1-\frac{1.81}{100})^t[/tex] and the percentage rate of decay is 1.81%.
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