Question

1. (20 points) you have 6.9 pounds of einsteinium-252, which has an exponential decay rate of 0.15% per day.

(a) find the exponential decay function.
ans:
(b) how much will be left after 1085 days? round your answer to the nearest tenth of a pound.
ans: pounds
(c) what is the half-life of einsteinium-252? round your answer to the nearest tenth of a day.
ans: days
(d) after how many days will there be 2.8 pounds left? round your answer to the nearest tenth of a day.

Explanation:

Step1: Identify initial values

$$A_0 = 6.9, \quad r = 0.15\% = 0.0015$$

Step2: Write decay function

$$A(t) = 6.9e^{-0.0015t}$$

Step3: Calculate amount remaining

$$A(1085) = 6.9e^{-0.0015 \times 1085} \approx 1.352$$

Step4: Round to nearest tenth

$$A(1085) \approx 1.4$$

Step5: Set up half-life equation

$$0.5 = e^{-0.0015t}$$

Step6: Solve for half-life

$$t = \frac{\ln(0.5)}{-0.0015} \approx 462.098$$

Step7: Round half-life value

$$t \approx 462.1$$

Step8: Set up target amount equation

$$2.8 = 6.9e^{-0.0015t}$$

Step9: Solve for target time

$$t = \frac{\ln(2.8 / 6.9)}{-0.0015} \approx 601.066$$

Step10: Round target time value

$$t \approx 601.1$$

Answer:

(a) \(A(t) = 6.9e^{-0.0015t}\)
(b) \(1.4\) pounds
(c) \(462.1\) days
(d) \(601.1\) days