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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?

1 Answer

5 votes

Answer:

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

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