Certainly! Let's solve this step-by-step:
The given function representing the mass remaining after [tex]\( t \)[/tex] years is:
[tex]\[ m(t) = 330e^{-0.03t} \][/tex]
where [tex]\( m(t) \)[/tex] is in grams.
### Part (a): Find the mass at time [tex]\( t = 0 \)[/tex]
To find the mass at time [tex]\( t = 0 \)[/tex], substitute [tex]\( t = 0 \)[/tex] into the given function:
[tex]\[ m(0) = 330e^{-0.03 \cdot 0} \][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[ m(0) = 330e^0 \][/tex]
[tex]\[ m(0) = 330 \times 1 \][/tex]
[tex]\[ m(0) = 330 \][/tex]
So, the mass at time [tex]\( t = 0 \)[/tex] is:
[tex]\[ \boxed{330.0 \text{ grams}} \][/tex]
### Part (b): How much of the mass remains after 20 years?
To determine the mass remaining after 20 years, substitute [tex]\( t = 20 \)[/tex] into the function:
[tex]\[ m(20) = 330e^{-0.03 \cdot 20} \][/tex]
Let's simplify the exponent first:
[tex]\[ -0.03 \cdot 20 = -0.6 \][/tex]
Now, substitute back into the equation:
[tex]\[ m(20) = 330e^{-0.6} \][/tex]
By evaluating [tex]\( e^{-0.6} \)[/tex] and then multiplying by 330, we find:
[tex]\[ m(20) \approx 181.1 \][/tex]
So, the mass after 20 years is:
[tex]\[ \boxed{181.1 \text{ grams}} \][/tex]
### Summary:
- (a) The mass at time [tex]\( t = 0 \)[/tex] is [tex]\( 330.0 \)[/tex] grams.
- (b) The mass remaining after 20 years is [tex]\( 181.1 \)[/tex] grams.
Both answers have been rounded to 1 decimal place as required.
