Answer:
The hourly decay rate is of 1.25%, so the hourly rate of change is of -1.25%.
The function to represent the mass of the sample after t days is [tex]A(t) = 810(0.74)^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.
Hourly rate of change:
Decreases 26% by day. A day has 24 hours. This means that [tex]A(24) = (1-0.26)A(0) = 0.74A(0)[/tex]; We use this to find r.
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]0.74A(0) = A(0)(1-r)^{24}[/tex]
[tex](1-r)^{24} = 0.74[/tex]
[tex]\sqrt[24]{(1-r)^{24}} = \sqrt[24]{0.74}[/tex]
[tex]1 - r = (0.74)^{\frac{1}{24}}[/tex]
[tex]1 - r = 0.9875[/tex]
[tex]r = 1 - 0.9875 = 0.0125[/tex]
The hourly decay rate is of 1.25%, so the hourly rate of change is of -1.25%.
Starts out with 810 grams of Element X
This means that [tex]A(0) = 810[/tex]
Element X is a radioactive isotope such that its mass decreases by 26% every day.
This means that we use, for this equation, r = 0.26.
The equation is:
[tex]A(t) = A(0)(1-r)^t[/tex]
[tex]A(t) = 810(1 - 0.26)^t[/tex]
[tex]A(t) = 810(0.74)^t[/tex]
The function to represent the mass of the sample after t days is [tex]A(t) = 810(0.74)^t[/tex]
