Question

The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 120I I2 + I + 4 where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum? I = thousand foot-candles

Answer 1

Answer:

I=2 thousand foot-candles

Step-by-step explanation:

P is maximum in one of   its critical points, to find the critical points of a function you have to find the first derivative and equal it to zero:

$P=\frac{120I}{I^2+I+4}$

$P'=\frac{(I^2+I+4)(120I)'-(120I)(I^2+I+4)'}{(I^2+I+4)^2}\\P'=\frac{(I^2+I+4)120-(120I)(2I+1)}{(I^2+I+4)^2}\\P'=\frac{120I^2+120I+480-240I^2-120I}{(I^2+I+4)^2}\\P'=\frac{-120I^2+480}{(I^2+I+4)^2}$

$P'=\frac{-120I^2+480}{(I^2+I+4)^2}=0\\\frac{-120I^2+480}{(I^2+I+4)^2}=0\\-120I^2+480=0\\-120(I^2-4)=0\\(I+2)(I-2)=0\\\therefore I_1=-2 , I_2=2$

The critical points are $I_1=-2 , I_2=2$, now evaluate this values in the function:

$P(-2)=\frac{120(-2)}{(-2)^2+(-2)+4}=\frac{-240}{4-2+4} =\frac{-240}{6}=-40$

$]P(2)=\frac{120(2)}{(2)^2+(2)+4}=\frac{240}{4+2+4} =\frac{240}{10}=24$

Therefore the rate P is at the maximum when I=2 thousand foot-candles