Answer to Question #99002 in Calculus for Kyla

Question

(a) Find expressions for a plant leaf epidermal cell’s absorption and consumption of a nutrient (as for the spherical cell in Chapter 1). Assume the absorption rate per unit surface area is k1 and the consumption rate per unit volume is k2.
(b) For what values of r is absorption faster than consumption? You answer should give a range of r values defined in terms of the other parameters given in the problem (h, k1, k2).

Answer 1

If a plant leaf epidermal cell is a cylinder with radius r and height h then it's volume is given by $V = \pi r^2h$ and surface $S = 2(\pi rh + \pi r^2).$

(a) Thus, absorption of a nutrient is $A = k_1S =2 k_1\pi(rh + r^2)$ and consumption is $C =k_2V= k_2h\pi r^2$ .

(b) Let's find r for whitch $A>C$.

$2 k_1\pi(rh + r^2)> k_2h\pi r^2\\ r^2(2k_1\pi-k_2\pi h)+2k_1\pi hr>0\\ r^2+\dfrac{2k_1h}{2k_1-k_2h}r>0$.

Solving this inequality and taking only positive radii obtain:

$r \in ( \dfrac{2k_1h}{k_2h-2k_1},+\infty)$ if $k_2>\dfrac{2k_1}{h}$ and

$r \in ( 0,+\infty)$ in another case.