Questions: Starting 90 is a racehorse means that the doors according to the function A(t) = A0 e^0.024t, where A0 is the initial amount present and t is time in years. Assume that a scientist has a sample of 500 grams. (a) What is the decay rate of technetium-99? (b) How much will 240 grams of technetium-99 be left? (c) What is its half-life, assuming? (d) When will only 200 grams of technetium remain? (e) What is the initial amount?

Solution Steps

The function given is \( A(t) = A_0 e^{0.024t} \). This represents exponential growth, not decay. Therefore, the decay rate is not applicable here. The rate given is a growth rate of 0.024 per year.

To find how much of a 240-gram sample remains, we use the formula: \[ A(t) = 240 e^{0.024t} \] However, without a specific time \( t \), we cannot calculate the remaining amount.

The half-life \( t_{1/2} \) is found using the formula: \[ t_{1/2} = \frac{\ln(2)}{k} \] where \( k = 0.024 \).

Calculating: \[ t_{1/2} = \frac{\ln(2)}{0.024} \approx 28.88 \text{ years} \]

Final Answer