Question

1. (20 points) you have (13.1) pounds of copernicium-285, which has an exponential decay rate of (2.39%) per second.

(a) find the exponential decay function.

ans:

(b) how much will be left after (87) seconds? round your answer to the nearest tenth of a pound.

ans: pounds

(c) what is the half-life of copernicium-285? round your answer to the nearest tenth of a second.

ans: seconds

(d) after how many seconds will there be (2.6) pounds left? round your answer to the nearest tenth of a second.

Explanation:

Step1: Identify the decay function parameters

$$A_0 = 13.1, \quad r = 0.0239$$

Step2: Write the exponential decay function

$$A(t) = 13.1e^{-0.0239t}$$

Step3: Calculate the amount after 87 seconds

$$A(87) = 13.1e^{-0.0239 \times 87} \approx 1.637$$

Step4: Round the amount to nearest tenth

$$A(87) \approx 1.6$$

Step5: Set up the half-life equation

$$e^{-0.0239t} = 0.5$$

Step6: Solve for the half-life time

$$t = \frac{\ln(0.5)}{-0.0239} \approx 29.002$$

Step7: Round the half-life to nearest tenth

$$t \approx 29.0$$

Step8: Set up equation for 2.6 pounds

$$13.1e^{-0.0239t} = 2.6$$

Step9: Solve for the time \(t\)

$$t = \frac{\ln(2.6 / 13.1)}{-0.0239} \approx 67.73$$

Step10: Round the time to nearest tenth

$$t \approx 67.7$$

Answer:

(a) \(A(t) = 13.1e^{-0.0239t}\)
(b) \(1.6\) pounds
(c) \(29.0\) seconds
(d) \(67.7\) seconds