Active oxygen and free radicals are believed to be exacerbating factors in causing cell injury and aging in living tissue (see citation below). Researchers are therefore interested in understanding the protective role of natural antioxidants. In the study of one such antioxidant (Hsian-tsao leaf gum), the antioxidation activity of the substance has been found to depend on concentration in the following way:

$$\frac{dA(c)}{dc} = k[L - A(c)], \ \ \ A(0) = 0.$$
In this equation, the dependent variable $A$ is a quantitative measure of antioxidant activity at concentration $c$. The constant $L$ represents a limiting or equilibrium value of this activity, and $k$ a positive rate constant.

1. Let $B(c) = A(c) - L$ and reformulate the given initial value problem in terms of this new dependent variable, $B$.
   
   $\displaystyle \frac{dB(c)}{dc} =$ help (formulas)
   
   $B(0) =$ help (formulas)
   
   
2. Solve the new initial value problem for $B(c)$ and then determine the quantity $A(c)$.
   
   $A(c) =$ help (formulas)
   
   Does the activity $A(c)$ ever exceed the value $L$? ? yes no
   
   
3. Determine the concentration at which 65% of the limiting antioxidation activity is achieved. (Your answer is a function of the rate constant $k$.)
   
   $c =$ help (numbers)

Lih-Shiuh, Su-Tze Chou, and Wen-Wan Chao, "Studies on the Antioxidative Activities of Hsian-tsao (Mesona procumbens Hemsl) Leaf Gum," J. Agric. Food Chem., Vol. 49, 2001, pp. 963-968.