Question

an exponential decay function can be used to model the number of grams of a radioactive material that remain after a period of time. a radioactive isotope decays over time, with the amount remaining after t years given by ( y = 200e^{-0.00012877t} ) if 200 grams is the original amount. (a) how much remains after 3500 years? (b) use graphical methods to estimate the number of years until 40 grams of a radioactive isotope remain. (a) the amount of a radioactive isotope remaining after 3500 years is approximately grams. (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Substitute \( t = 3500 \) into the formula

We have the formula \( y=200e^{-0.0001287t} \). Substitute \( t = 3500 \) into it: \( y = 200e^{-0.0001287\times3500} \).
First, calculate the exponent: \( - 0.0001287\times3500=-0.45045 \).

Step2: Calculate the value of \( e^{-0.45045} \)

We know that \( e^{-x}=\frac{1}{e^{x}} \). Using a calculator, \( e^{-0.45045}\approx0.6376 \).

Step3: Calculate the value of \( y \)

Then \( y = 200\times0.6376 = 127.52\approx127.5 \)

Answer:

\( 127.5 \)