Certainly! Let's go through the problem step-by-step.
### Part (a): Finding the mass at time [tex]\( t = 0 \)[/tex]
The function that describes the mass remaining after [tex]\( t \)[/tex] days is given by:
[tex]\[ m(t) = 14 e^{-0.017 t} \][/tex]
To find the mass at time [tex]\( t = 0 \)[/tex], we simply substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[ m(0) = 14 e^{-0.017 \cdot 0} \][/tex]
Since [tex]\( -0.017 \cdot 0 = 0 \)[/tex], we have:
[tex]\[ m(0) = 14 e^0 \][/tex]
Recall that [tex]\( e^0 = 1 \)[/tex], so:
[tex]\[ m(0) = 14 \times 1 = 14 \, \text{kg} \][/tex]
Thus, the mass at [tex]\( t = 0 \)[/tex] is:
[tex]\[ 14 \, \text{kg} \][/tex]
### Part (b): Finding the mass after 42 days
Next, we need to determine how much mass remains after 42 days. We'll use the function [tex]\( m(t) \)[/tex] and substitute [tex]\( t = 42 \)[/tex].
So, we need to find [tex]\( m(42) \)[/tex]:
[tex]\[ m(42) = 14 e^{-0.017 \cdot 42} \][/tex]
First, we compute the exponent:
[tex]\[ -0.017 \cdot 42 = -0.714 \][/tex]
Now, we calculate the value of [tex]\( e^{-0.714} \)[/tex]. Utilizing the fact that we've determined this part already, we find the remaining mass is:
[tex]\[ m(42) = 14 e^{-0.714} \approx 6.9 \, \text{kg} \][/tex]
To conclude, the mass remaining after 42 days, rounded to one decimal place, is:
[tex]\[ 6.9 \, \text{kg} \][/tex]
Therefore, the answers are:
(a) [tex]\( 14 \, \text{kg} \)[/tex]
(b) [tex]\( 6.9 \, \text{kg} \)[/tex]
