Sure, let's solve this step by step.
(a) To find the mass at time [tex]\( t = 0 \)[/tex]:
The given function for mass is:
[tex]\[
m(t) = 13e^{-0.017t}
\][/tex]
Plug in [tex]\( t = 0 \)[/tex] into the function:
[tex]\[
m(0) = 13e^{-0.017 \cdot 0}
\][/tex]
Simplify the exponent:
[tex]\[
m(0) = 13e^0
\][/tex]
Since [tex]\( e^0 = 1 \)[/tex], we have:
[tex]\[
m(0) = 13 \cdot 1 = 13
\][/tex]
Therefore, the mass at time [tex]\( t = 0 \)[/tex] is:
[tex]\[
13 \, \text{kg}
\][/tex]
(b) To find the mass remaining after 42 days (rounded to one decimal place):
Using the same function for mass:
[tex]\[
m(t) = 13e^{-0.017t}
\][/tex]
Plug in [tex]\( t = 42 \)[/tex]:
[tex]\[
m(42) = 13e^{-0.017 \cdot 42}
\][/tex]
Calculate the exponent:
[tex]\[
m(42) = 13e^{-0.714}
\][/tex]
Evaluating [tex]\( e^{-0.714} \)[/tex] and multiplying by 13 gives:
[tex]\[
m(42) \approx 6.4
\][/tex]
Therefore, the mass remaining after 42 days, rounded to one decimal place, is:
[tex]\[
6.4 \, \text{kg}
\][/tex]
So, the answers are:
(a) The mass at time [tex]\( t = 0 \)[/tex] is [tex]\( 13 \)[/tex] kg.
(b) After 42 days, the mass remaining is approximately [tex]\( 6.4 \)[/tex] kg.
