Question

element X is a radioactive isotope such that every 21 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 80 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 3.25%.

The function showing the mass of the sample remaining after t is
`A(t) = 80(0.9675)^t`

Explanation:

Equation for decay of substance:

The equation that models the amount of a decaying substance after t years 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 21 years, its mass decreases by half.

This means that
`A(21) = 0.5A(0)`. We use this to find r, the percentage rate of decay per year.

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

`0.5A(0) = A(0)(1-r)^(21)`

`(1-r)^(21) = 0.5`

`\sqrt[21]{(1-r)^(21)} = \sqrt[21]{0.5}`

`1 - r = 0.5^{(1)/(21)}`

`1 - r = 0.9675`

`r = 1 - 0.9675 = 0.0325`

The percentage rate of decay per year is of 3.25%.

Given that the initial mass of a sample of Element X is 80 grams.

This means that
`A(0) = 80`

The equation is:

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

`A(t) = 80(1-0.0325)^t`

`A(t) = 80(0.9675)^t`

The function showing the mass of the sample remaining after t is
`A(t) = 80(0.9675)^t`