Question

The continuous growth and decay formulas are given below.

Continuous Growth: A = Peʳᵗ
Continuous Decay: A = Pe⁻ᵗ
A = amount at time t
P = initial amount
r = rate of growth or decay
t = time
Plutonium 241 has a decay rate of 4.8% per year.
How many years will it take a 50 kg sample to decay to 10 kg?

Answer 1

To find the number of years it will take for a 50 kg sample of Plutonium 241 to decay to 10 kg, we can use the continuous decay formula: A = Pe⁻ᵗ. Substituting the given values and solving for t, we find that it will take approximately 16.95 years.

Step-by-step explanation:

To find the number of years it will take for a 50 kg sample of Plutonium 241 to decay to 10 kg, we can use the continuous decay formula: A = Pe⁻ᵗ. We know the initial amount P is 50 kg, the amount remaining A is 10 kg, and the rate of decay r is 4.8% per year. We need to solve for t, the time it takes for the sample to decay.

Substituting the given values into the formula, we have 10 = 50e⁻⁰·⁰⁴⁸·ᵗ. To solve for t, we can take the natural logarithm (ln) of both sides: ln(10/50) = -0.048t. Simplifying further, ln(0.2) = -0.048t.

Now, we can divide both sides of the equation by -0.048 to isolate t: t = ln(0.2)/-0.048. Using a calculator, we find t ≈ 16.95 years. Therefore, it will take approximately 16.95 years for a 50 kg sample of Plutonium 241 to decay to 10 kg.