Find the​ half-life of a radioactive​ element, which decays according to the function ​A(t)=A0e−0.0303t​, where t is the time in years.

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Part 1
The​ half-life of the element is
enter your response here years.
​(Round to the nearest​ tenth.)

1 answer

To find the half-life of the element, we need to find the value of t when A(t) is equal to half of A0.

A(t) = A0e^(-0.0303t)

Setting A(t) = (1/2)A0:

(1/2)A0 = A0e^(-0.0303t)

Dividing both sides by A0:

(1/2) = e^(-0.0303t)

Taking the natural logarithm of both sides:

ln(1/2) = -0.0303t

Using a calculator to approximate ln(1/2) ≈ -0.693:

-0.693 = -0.0303t

Dividing both sides by -0.0303:

t ≈ 22.88

Therefore, the half-life of the radioactive element is approximately 22.9 years.