Question

Element X is a radioactive isotope such that every 25 years, its mass decreases by

Answer 1

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:

$A(t) = A(0)(0.5)^{\frac{t}{25}}$

In which A(0) is the initial amount.

Initial mass of a sample of Element X is 910 grams.

This means that $A(0) = 910$

So

$A(t) = A(0)(0.5)^{\frac{t}{25}}$

$A(t) = 910(0.5)^{\frac{t}{25}}$

How long would it be until the mass of the sample reached 400 grams?

This is t for which $A(t) = 400$

So

$A(t) = 910(0.5)^{\frac{t}{25}}$

$400 = 910(0.5)^{\frac{t}{25}}$

$(0.5)^{\frac{t}{25}} = \frac{400}{910}$

$(0.5)^{\frac{t}{25}} = 0.43956$

$\log{(0.5)^{\frac{t}{25}}} = \log{0.43956}$

$\frac{t}{25}\log{0.5} = \log{0.43956}$

$t = \frac{25\log{0.43956}}{\log{0.5}}$

$t = 25.6$

It will take 25.6 years for the mass of the sample to reach 400 grams.

Answer 2

Answer:

29.6

Step-by-step explanation: