The rate of accumulation of a specific enzyme in a bioreactor, R(t)R(t)R(t) in mmol/L per hour, is modeled by the function:
R(t)=7t−2(t+3)2,t≥0 R(t) = \frac{7t - 2}{(t + 3)^2}, \quad t \ge 0 R(t)=(t+3)27t−2,t≥0where t t\,t is the time in hours since the start of the reaction.
Express R(t)R(t)R(t) in the form At+3+B(t+3)2\displaystyle \frac{A}{t + 3} + \frac{B}{(t + 3)^2}t+3A+(t+3)2B, where A A\,A and B B\,B are constants to be found.
The total accumulation of the enzyme, KKK, between t=1t = 1t=1 and t=5t = 5t=5 hours is given by K=∫15R(t) dtK = \int_{1}^{5} R(t) \, dtK=∫15R(t)dt.
Show that K=p+lnqK = p + \ln qK=p+lnq, where p p\,p and q q\,q are rational numbers to be determined.
