Alright, let's solve the problem step-by-step using the given exponential decay function:
Given the decay function:
[tex]\[ m(t) = 405 e^{-0.03 t} \][/tex]
where [tex]\( m(t) \)[/tex] is the mass remaining after [tex]\( t \)[/tex] years.
### Part (a)
To find the mass at time [tex]\( t = 0 \)[/tex]:
1. Substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[ m(0) = 405 e^{-0.03 \cdot 0} \][/tex]
2. Simplify the exponent:
[tex]\[ m(0) = 405 e^{0} \][/tex]
3. Recall that [tex]\( e^0 = 1 \)[/tex]:
[tex]\[ m(0) = 405 \cdot 1 \][/tex]
4. Hence:
[tex]\[ m(0) = 405 \][/tex]
So, the mass at time [tex]\( t = 0 \)[/tex] is:
[tex]\[ \boxed{405.0} \text{ grams} \][/tex]
### Part (b)
To find the mass remaining after 10 years:
1. Substitute [tex]\( t = 10 \)[/tex] into the function:
[tex]\[ m(10) = 405 e^{-0.03 \cdot 10} \][/tex]
2. Simplify the exponent:
[tex]\[ m(10) = 405 e^{-0.3} \][/tex]
3. Use a calculator to find the value of [tex]\( e^{-0.3} \)[/tex]:
4. Multiply the base mass by this value:
[tex]\[ m(10) \approx 405 \cdot 0.7408 = 300 \][/tex]
So, the mass remaining after 10 years is:
[tex]\[ \boxed{300.0} \text{ grams} \][/tex]
In summary:
- The mass at time [tex]\( t = 0 \)[/tex] is [tex]\( 405.0 \)[/tex] grams.
- The mass remaining after 10 years is [tex]\( 300.0 \)[/tex] grams.
