A researcher determines that a certain protein with mass mt

A researcher determines that a certain protein with mass m(t) grams at time r(hrs) disintegrates into amino acids at a rate jointly proportional to m (t) and t. Experiments indicate that half of any given sample will distintegrate in 12 hrs.

(a) Set up and solve a differntial euqation for m (t).

(b) what function of a give sample remains after 9 hrs?

Solution

a) at time t

dm/dt = - ktm(t)

=> dm/m(t) = -ktdt

on integrating

m(t) = Mexp(-kt²/2) [where M is the initial mass of m(t)]

given in 12 hrs m(t) = M/2

=> k = 9.627 x 10^-3 /s²

So with the value of k differential equation and it\'s solution is there.

b)

at t = 9 hrs

m(t) = M exp( - 0.3899) = 0.677M

hence 67.7% fraction remains of M at t = 9 hrs