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

Answer:

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

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

Step-by-step 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.

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

$0.5A(0) = A(0)(1-r)^{21}$

$(1-r)^{21} = 0.5$

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

$1 - r = 0.5^{\frac{1}{21}}$

$1 - r = 0.9675$

$r = 1 - 0.9675 = 0.0325$

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

Answer 2

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