Let's go through each part step by step.

(a) Writing the decay function $P(t)$

Since strontium-90 decays at a rate of 2% per year, we use the exponential decay formula:

$P(t) = P_0 e^{-kt}$

where:

• $P_0 = 1$ (since at $t = 0$, 100% remains),
• $k$ is the decay constant.

Since the decay rate is 2% per year, we set $k = 0.02$, so:

$P(t) = e^{-0.02t}$

(b) Estimating the Half-Life

The half-life $T_{1/2}$ is found using the formula:

$T_{1/2} = \frac{\ln(2)}{k}$

Substituting $k = 0.02$:

$T_{1/2} = \frac{\ln(2)}{0.02} \approx \frac{0.693}{0.02} = 34.65 \text{ years}$

So, the estimated half-life is about 34.65 years.

(c) Finding the remaining mass after 30 years

Given an initial amount of 8.5 grams, we use:

$P(30) = 8.5 e^{-0.02(30)}$

Calculating the exponent:

$-0.02 \times 30 = -0.6$

$e^{-0.6} \approx 0.5488$

$P(30) = 8.5 \times 0.5488 \approx 4.66 \text{ grams}$

So, after 30 years, about 4.66 grams of strontium-90 remains.

Do you want more details or have any questions?

Here are 5 related questions to explore further:

1. How does the decay rate affect the half-life?
2. What is the general formula for exponential decay?
3. How would the function change if the decay rate was 5% instead?
4. How long would it take for only 1 gram of strontium-90 to remain?
5. Can we use logarithms to find the time for a specific remaining fraction?

Tip:

The half-life formula $T_{1/2} = \frac{\ln(2)}{k}$ is useful for any exponential decay problem!