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 127.5 grams. (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

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

Given \( y = 200e^{-0.00012877t} \), when \( t = 3500 \), we have \( y=200e^{-0.00012877\times3500} \).
First, calculate the exponent: \( - 0.00012877\times3500=-0.450695 \).

Step2: Calculate the value of \( e^{-0.450695} \)

Using the property that \( e^x \) (where \( x=-0.450695 \)), and \( e^{-0.450695}=\frac{1}{e^{0.450695}}\approx0.6375 \).

Step3: Calculate the value of \( y \)

Then \( y = 200\times0.6375 = 127.5 \).

Answer:

The amount of a radioactive isotope remaining after 3500 years is approximately \( 127.5 \) grams.