contestada

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?

Respuesta :

Answer:

At light intensity I = 3, is P a maximum

Explanation:

Given:

[tex]P=\frac{90I}{I^2+I+9}[/tex]

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

[tex]\frac{dp}{dI}=\frac{(I^2+I+9).\frac{d(90I)}{dI}-90I.\frac{d((I^2+I+9)}{dI}}{(I^2+I+9)^2}[/tex]

or

[tex]\frac{dp}{dI}=\frac{(I^2+I+9).90-90I.(2I+1)}{(I^2+I+9)^2}[/tex]

or

[tex]\frac{dp}{dI}=\frac{90I^2+90I+810)-(180I^2+90I)}{(I^2+I+9)^2}[/tex]

or

[tex]\frac{dp}{dI}=\frac{-90I^2+810)}{(I^2+I+9)^2}[/tex]

Now for the maxima [tex]\frac{dP}{dI}=0[/tex]

thus,

[tex]0=\frac{-90I^2+810)}{(I^2+I+9)^2}[/tex]

or

[tex]-90I^2+810=0[/tex]

or

[tex]I^2=\frac{810}{90}[/tex]

or

[tex]I^2=9[/tex]

or

I = 3

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

at I = 3

[tex]P=\frac{90\times3}{3^2+3+9}[/tex]

or

[tex]P=\frac{270}{21}[/tex]

or

[tex]P=12.85[/tex]

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