The concentration C C\,C of a chemical catalyst in a specialized solvent depends on the depth ddd (in cm) according to the model
C=502(3d−k),d≠k3 C = \frac{50}{2(3d - k)}, \quad d \neq \frac{k}{3} C=2(3d−k)50,d=3kwhere k k\,k is a positive constant and k≠12k \neq 12k=12.
Find dCdd\displaystyle \frac{dC}{dd}dddC giving your answer in simplest form in terms of kkk.
The point P P\,P with ddd-coordinate 4 lies on the curve. Given that the rate of change of concentration with respect to depth at P P\,P is -3,
find the two possible values of kkk.
Given also that k<12k < 12k<12,
find the equation of the normal to the curve at PPP, writing your answer in the form ad+bC+c=0ad + bC + c = 0ad+bC+c=0, where aaa, bbb, and c c\,c are integers to be found.
Show, using algebraic integration, that
∫35502(3d−k) dd=λln2 \int_{3}^{5} \frac{50}{2(3d - k)} \, dd = \lambda \ln 2 ∫352(3d−k)50dd=λln2where λ \lambda\,λ is a constant to be found.
