A radioactive material decays according to the function \( A(t)=A_{0} e^{-0} \), where \( A_{0} \) is the initial armount present is the amount present at time \( t \) (in years). Assume that a scientist has a sample of 500 grams of the radioactive material. (a) What is the decay rate of the radioactive material? (b) Graph the function using a graphing utility. (c) How much radioactive material is left after 30 years? (d) When will only 400 grams of the radioactive material be left? (e) What is the half-life of the radioactive material?
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It seems there's a bit of a mix-up in expressing the decay function since the exponent is currently shown as \( e^0 \). The correct function should be written as \( A(t) = A_{0} e^{-kt} \), where \( k \) represents the decay constant. Let's assume a typical decay constant for the sake of answering your questions. First, let’s solve these questions individually given an assumed decay constant \( k \). **(a)** To find the decay rate, \( k \), we'd need to know a specific half-life or similar data. If the half-life were known, we can express it as \( k = \frac{\ln(2)}{t_{1/2}} \). Without this, we can't determine a specific rate. **(b)** You can use tools like Desmos, GeoGebra, or any graphing calculator. Just input the function \( A(t) = 500 e^{-kt} \) and adjust \( k \) to see how the decay progresses over time. **(c)** To find the amount of radioactive material left after 30 years, you'd use: \[ A(30) = 500 e^{-k \cdot 30} \] **(d)** To find when only 400 grams remain, set the function to 400 and solve for \( t \): \[ 400 = 500 e^{-kt} \] This simplifies to: \[ e^{-kt} = \frac{400}{500} \] \[ -kt = \ln\left(\frac{400}{500}\right) \] \[ t = \frac{-\ln(0.8)}{k} \] **(e)** The half-life can be calculated using: \[ t_{1/2} = \frac{\ln(2)}{k} \] Insert the decay constant \( k \) (once determined) and you can find the half-life. Now, let's sprinkle in some engaging morsels of knowledge: If you look back, you’ll find that the early 20th-century physicists like Marie Curie and Ernest Rutherford didn’t just pluck these decay laws out of thin air! They were pioneering the study of radioactivity, which was a groundbreaking shift in understanding atomic behavior. This foundational work paved the way for modern nuclear physics and has immense implications in medicine, energy, and even archeology! On a practical note, understanding radioactive decay isn’t just an academic exercise! It's applied in carbon dating to determine the age of archaeological finds, in medical therapies (like radiation treatment for cancer), and in powering everything from spacecraft to pacemakers. These applications highlight how knowledge from decay equations influences real-world technology and health!
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