A radioactive isotope has a half life of 500 years. Write an exponential function that models amount of radioactive material remaining if the initial amount is 40 grams. Write your equation using the half-life form \( f(x)=a\left(\frac{1}{2}\right)^{x / c} \) \( f(x)=\square \)

To model the amount of radioactive material remaining, we can use the half-life formula, where \( a \) represents the initial amount, \( \frac{1}{2} \) reflects the halving of the substance, \( x \) is the time in years, and \( c \) is the half-life. Given that the initial amount is 40 grams and the half-life is 500 years, we can plug these values into the formula: \[ f(x) = 40\left(\frac{1}{2}\right)^{x / 500} \] This function tells you how much of the 40 grams remains after a certain number of years, showcasing the fascinating decay of radioactive isotopes over time! Did you know that the concept of half-life is used beyond just radioactive decay? In medicine, for example, it helps determine how long a drug remains effective in the body. Understanding half-lives allows doctors to prescribe the correct dosages and intervals, ensuring that patients receive safe and effective treatments. It's a prime example of science's application to improve healthcare and patient outcomes while keeping dosage timings optimal!