Question

The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 90I I2 + I + 9 where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?

Answer 1

At light intensity I = 3, is P a maximum

Step-by-step explanation:

Given:

`P=(90I)/(I^2+I+9)`

now differentiating the above equation with respect to Intensity 'I' we get

`(dp)/(dI)=((I^2+I+9).(d(90I))/(dI)-90I.(d((I^2+I+9))/(dI))/((I^2+I+9)^2)`

or

`(dp)/(dI)=((I^2+I+9).90-90I.(2I+1))/((I^2+I+9)^2)`

or

`(dp)/(dI)=(90I^2+90I+810)-(180I^2+90I))/((I^2+I+9)^2)`

or

`(dp)/(dI)=(-90I^2+810))/((I^2+I+9)^2)`

Now for the maxima
`(dP)/(dI)=0`

thus,

`0=(-90I^2+810))/((I^2+I+9)^2)`

or

`-90I^2+810=0`

or

`I^2=(810)/(90)`

or

`I^2=9`

or

I = 3

thus, for the value of intensity I = 3, the P is maximum

at I = 3

`P=(90*3)/(3^2+3+9)`

or

`P=(270)/(21)`

or

`P=12.85`