The amount of a radioactive isotope present at time t is given by Upper A left parenthesis t right parenthesis equals 800 e Superscript negative 0.02825 t ​grams, where t is the time in years that the isotope decays. The initial amount present is 800 grams. Complete parts ​(a) through ​(c).

Question

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Part 1
​(a) How many grams remain after 15 ​years?
  
enter your response here g​ (Type an integer or decimal rounded to two decimal places as​ needed.)

Answer 1

To calculate the amount of the radioactive isotope remaining after 15 years, we can use the given formula:

\[ A(t) = 800 e^{-0.02825 t} \]

Substituting \( t = 15 \):

\[ A(15) = 800 e^{-0.02825 \times 15} \]

First, calculate the exponent:

\[ -0.02825 \times 15 = -0.42375 \]

Now calculate \( e^{-0.42375} \):

\[ e^{-0.42375} \approx 0.65458 \]

Now substitute this back into the equation for \( A(15) \):

\[ A(15) = 800 \times 0.65458 \approx 523.664 \]

Rounding to two decimal places, the amount remaining after 15 years is approximately:

\[ A(15) \approx 523.66 \text{ grams} \]

So the answer is:

523.66 grams