Question
Radium-226 decays in such a way that after
years, the amount, in grams, remaining can be modeled exponentially.
An initial amount of
grams of Radium-226 can be modeled by this function.
A(t)=450e−0.0004t
How many grams of Radium-226 will remain after 7
years? Round your answer to the nearest tenth of a gram.
(1 point)
Answers
GPT-4o mini
To find the amount of Radium-226 remaining after 7 years, we will substitute \( t = 7 \) into the function \( A(t) \).
The function given is:
\[
A(t) = 450 e^{-0.0004t}
\]
Substituting \( t = 7 \):
\[
A(7) = 450 e^{-0.0004 \times 7}
\]
Calculating \( -0.0004 \times 7 \):
\[
-0.0004 \times 7 = -0.0028
\]
Now we can rewrite the equation:
\[
A(7) = 450 e^{-0.0028}
\]
Next, we need to calculate \( e^{-0.0028} \). Using a calculator or an approximation, we find:
\[
e^{-0.0028} \approx 0.9972
\]
So now substitute this value back into the equation:
\[
A(7) = 450 \times 0.9972
\]
Calculating that gives:
\[
A(7) \approx 448.74
\]
Now, round to the nearest tenth of a gram:
\[
\boxed{448.7}
\]
Therefore, after 7 years, approximately 448.7 grams of Radium-226 will remain.
The function given is:
\[
A(t) = 450 e^{-0.0004t}
\]
Substituting \( t = 7 \):
\[
A(7) = 450 e^{-0.0004 \times 7}
\]
Calculating \( -0.0004 \times 7 \):
\[
-0.0004 \times 7 = -0.0028
\]
Now we can rewrite the equation:
\[
A(7) = 450 e^{-0.0028}
\]
Next, we need to calculate \( e^{-0.0028} \). Using a calculator or an approximation, we find:
\[
e^{-0.0028} \approx 0.9972
\]
So now substitute this value back into the equation:
\[
A(7) = 450 \times 0.9972
\]
Calculating that gives:
\[
A(7) \approx 448.74
\]
Now, round to the nearest tenth of a gram:
\[
\boxed{448.7}
\]
Therefore, after 7 years, approximately 448.7 grams of Radium-226 will remain.