Answer:
It will take 25.6 years for the mass of the sample to reach 400 grams.
Step-by-step explanation:
Element X is a radioactive isotope such that every 25 years, its mass decreases by half.
This means that the amount of the substance after t years can be modeled by a function in the following format:
[tex]A(t) = A(0)(0.5)^{\frac{t}{25}}[/tex]
In which A(0) is the initial amount.
Initial mass of a sample of Element X is 910 grams.
This means that [tex]A(0) = 910[/tex]
So
[tex]A(t) = A(0)(0.5)^{\frac{t}{25}}[/tex]
[tex]A(t) = 910(0.5)^{\frac{t}{25}}[/tex]
How long would it be until the mass of the sample reached 400 grams?
This is t for which [tex]A(t) = 400[/tex]
So
[tex]A(t) = 910(0.5)^{\frac{t}{25}}[/tex]
[tex]400 = 910(0.5)^{\frac{t}{25}}[/tex]
[tex](0.5)^{\frac{t}{25}} = \frac{400}{910}[/tex]
[tex](0.5)^{\frac{t}{25}} = 0.43956[/tex]
[tex]\log{(0.5)^{\frac{t}{25}}} = \log{0.43956}[/tex]
[tex]\frac{t}{25}\log{0.5} = \log{0.43956}[/tex]
[tex]t = \frac{25\log{0.43956}}{\log{0.5}}[/tex]
[tex]t = 25.6[/tex]
It will take 25.6 years for the mass of the sample to reach 400 grams.
