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.)
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.
To find the half-life of the radioactive element, we need to find the value of t when A(t) is half of its initial value, A₀.
A(t) = A₀e^(-0.0303t)
Since we are looking for the half-life, we set A(t) equal to half of A₀:
A(t) = (1/2)A₀
(1/2)A₀ = A₀e^(-0.0303t)
Now, let's solve for t. Divide both sides of the equation by A₀:
(1/2) = e^(-0.0303t)
Next, take the natural logarithm (ln) of both sides of the equation to eliminate the exponential function:
ln(1/2) = ln(e^(-0.0303t))
The natural logarithm of e raised to any power t is simply t, so we have:
ln(1/2) = -0.0303t
Now, divide both sides of the equation by -0.0303:
t = ln(1/2) / -0.0303
Using a calculator, we can evaluate this:
t ≈ 22.8
Therefore, the half-life of the radioactive element is approximately 22.8 years.
To find the half-life of a radioactive element that follows the function A(t) = A0e^(-0.0303t), we need to find the time (t) at which A(t) is equal to half of the initial amount (A0).
The initial amount of the radioactive element is A0, which is the value of A(t) at t = 0. In this case, A(t) = A0e^(-0.0303t). Therefore, when t = 0, A(0) = A0e^(-0.0303*0) = A0e^(0) = A0.
Now, let's find the time t at which A(t) is equal to half of the initial amount (A0/2):
A(t) = A0e^(-0.0303t)
A0/2 = A0e^(-0.0303t)
We can simplify this equation by dividing both sides by A0:
1/2 = e^(-0.0303t)
To solve for t, we need to take the natural logarithm of both sides:
ln(1/2) = ln(e^(-0.0303t))
ln(1/2) = -0.0303t
Now, divide both sides by -0.0303:
t = ln(1/2) / -0.0303
Using a calculator, evaluate ln(1/2) / -0.0303, and round the answer to the nearest tenth. This will give you the half-life of the radioactive element in years.