The amount of a radioactive isotope present at time t is given by Upper A left parenthesis t right parenthesis equals 400 e Superscript negative 0.02833 t ​grams, where t is the time in years that the isotope decays. The initial amount present is 400 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 find the amount of the radioactive isotope remaining after 15 years, we can use the formula provided:

\[ A(t) = 400 e^{-0.02833 t} \]

Substituting \( t = 15 \) into the equation:

\[ A(15) = 400 e^{-0.02833 \times 15} \]

First, calculate the exponent:

\[ -0.02833 \times 15 = -0.42495 \]

Now substitute this back into the equation:

\[ A(15) = 400 e^{-0.42495} \]

Next, we can calculate \( e^{-0.42495} \):

\[ e^{-0.42495} \approx 0.6533 \quad (\text{using a scientific calculator}) \]

Now, we can continue:

\[ A(15) = 400 \times 0.6533 \approx 261.32 \]

So, the amount of the radioactive isotope remaining after 15 years is approximately:

\[ \boxed{261.32} \text{ grams} \]