Hi,
It is possible to proceed if we try to analyse the
problem statement carefully.
1. "decay at a rate proportional to the amount present"
means:
dm/dt = km, where k is the
constant of proportionality.
2. "having mass of 100g originally" means:
m = 100 when t = 0.
3. "After 40 hours, its mass reduces to 80g" means:
m = 80 when t = 40.
In part (a), we have dm/dt = km.
In part (b), we solve the differential equation by the
method of variable separable (i.e put all the m's on one side and
all the t's on the other).
integral 1/m dm = integral k dt
ln |m| = kt + c
|m| = e^(kt + c) = A e^(kt), where A = e^c
m = plus/minus A e^(kt)
m = B e^(kt), where B = plus/minus A.
Now, we will use the information in points
2 and 3 to solve for B and then k, which I'll leave you to
continue. Answers: B = 100, t = 1/80 ln (4/5).
In (c), just find m when t = 80.
In (d), apply the same analysis as in (a).
Thanks!
Cheers,
Wen Shih
