1. Understanding Exponential Decay
Exponential decay follows the formula:
* A(t) = A₀ * e^(-kt)
where:
* A(t) is the amount remaining after time 't'
* A₀ is the initial amount
* k is the decay constant
* e is the base of the natural logarithm (approximately 2.718)
2. Finding the Decay Constant (k)
* Half-life: The time it takes for half of the radioactive material to decay.
* Relationship: We know that when t = half-life (75 days), A(t) = A₀/2. Let's substitute this into the formula:
A₀/2 = A₀ * e^(-k * 75)
Divide both sides by A₀:
1/2 = e^(-75k)
Take the natural logarithm of both sides:
ln(1/2) = -75k
Solve for k:
k = -ln(1/2) / 75 ≈ 0.00924
3. The Exponential Function
Now that we know the decay constant, we can write the function:
* A(t) = 381 * e^(-0.00924t)
4. Finding the Remaining Mass After a Given Time
To find the amount remaining after a specific time, simply substitute the time 't' into the function. For example, to find the amount remaining after 150 days:
* A(150) = 381 * e^(-0.00924 * 150) ≈ 95.25 kg
Therefore, the exponential function that models the decay is A(t) = 381 * e^(-0.00924t), and after 150 days, approximately 95.25 kg of the radioactive material will remain.
