To determine the mass remaining after 50 years for a substance that decays according to the function [tex]\( m(t) = 120 e^{-0.018 t} \)[/tex]:
1. Identify the initial parameters:
- The initial mass ([tex]\(m(0)\)[/tex]) is 120 kg.
- The decay constant is 0.018.
- Time [tex]\( t \)[/tex] is 50 years.
2. Substitute the time into the function:
[tex]\[
m(50) = 120 e^{-0.018 \times 50}
\][/tex]
3. Simplify the exponentiation:
[tex]\[
-0.018 \times 50 = -0.9
\][/tex]
4. Substitute the exponent back into the equation:
[tex]\[
m(50) = 120 e^{-0.9}
\][/tex]
5. Calculate the value of the exponential expression [tex]\( e^{-0.9} \)[/tex].
- Using a scientific calculator or appropriate software, you find:
[tex]\[
e^{-0.9} \approx 0.4065696597
\][/tex]
6. Multiply this result by the initial mass:
[tex]\[
m(50) = 120 \times 0.4065696597 \approx 48.7883591688719
\][/tex]
7. Round the result to two decimal places:
[tex]\[
m(50) \approx 48.79
\][/tex]
Therefore, the mass remaining after 50 years is approximately 48.79 kilograms.
