Photosynthesis. Find simplified formulas (depending on a, b, and c) for the following:

Question

 Photosynthesis. The rate at which photosynthesis takes place for a species of phytoplankton is modeled by the function below, where P is the rate of photosynthesis, I is the intensity of light, and the parameters a, b, and c are positive constants.

Find simplified formulas (depending on a, b, and c) for the following:

 

a) The initial slope,

          

 b) The light intensity for which P reaches a maximum:

c) The maximum rate of photosynthesis (maximum value of P):

Answer 1

Hello ALi,

a) Find The initial slope,

You need to find P'(l), the derivative of the function P. You will use the quotient rule for that. which is:

(u/v)' = [v*u' - u*v']/v²

Let u = numerator = l; then u' = 1

Let v = denominator = al² + bl + c; then v' = 2al + b

P' = [1(al² + bl + c) - l(2al + b)]/(al² + bl + c)² = (al² + bl + c-2al² -bl)/(al² + bl + c)²

Simplifying the numerator: P' = (-al² + c)/(al² + bl + c)²

Now to find P'(0), replace any mention of l by 0:

P'(0) = (-a*0² + c)/(a*0² + b*0 + c)²

so P'(0) = c/(c)²

P'(0) = c/c² ---> P'(0) = 1/c

b)The light intensity for which P reaches a maximum: You would need to do the 1st derivative test:

P'(l) = 0 ----> (-al² + c)/(al² + bl + c)²=0

Multiply both sides by (al² + bl + c)² : (-al² + c)=0 ----> c= al²

so l^2=c/a ---> l = ± square root(c/a)

Since: c and a are POSITIVE constants and the light intensity cannot be negative, you would only choose the positive square root ---> l = square root(c/a)

That will be the light intensity when P reaches its maximum.

c) The maximum rate of photosynthesis (maximum value of P): From the previous question, you know that the maximum value of P happens when l = square root of (c/a).

The equation of P you listed was: P = l/(al² + bl + c)

To find the max value of P, you would replace any mention of l by square root(c/a) and simplify where necessary

Cheers!