Question

Element X is a radioactive isotope such that every 13 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 520 grams, write a function showing the mass of the sample remaining after tt years, where the annual decay rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of decay per year, to the nearest hundredth of a percent.

Answer 1

The percentage rate of decay per year is of 5.19%.

The function showing the mass of the sample remaining after t is
`A(t) = 520(0.9481)^t`

Explanation:

Exponential equation of decay:

The exponential equation for the amount of a substance is given by:

`A(t) = A(0)(1-r)^t`

In which A(0) is the initial amount and r is the decay rate, as a decimal.

Every 13 years, its mass decreases by half.

This means that
`A(13) = 0.5A(0)`. We use this to find r. So

`A(t) = A(0)(1-r)^t`

`0.5A(0) = A(0)(1-r)^(13)`

`(1-r)^(13) = 0.5`

`\sqrt[13]{(1-r)^(13)} = \sqrt[13]{0.5}`

`1 - r = (0.5)^(1)/(13)`

`1 - r = 0.9481`

`r = 1 - 0.9481`

`r = 0.0519`

The percentage rate of decay per year is of 5.19%.

The initial mass of a sample of Element X is 520 grams

This means that
`A(0) = 520`. So

`A(t) = A(0)(1-r)^t`

`A(t) = 520(1-0.0519)^t`

`A(t) = 520(0.9481)^t`

The function showing the mass of the sample remaining after t is
`A(t) = 520(0.9481)^t`