The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function
P =
110I
I2 + I + 4
where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?
To find the light intensity at which the photosynthesis rate (P) is a maximum, we need to find the critical points of the function. In calculus, a critical point occurs where the derivative of the function is either zero or undefined.
1. Start by finding the derivative of the function P with respect to I. Let's call the derivative dP/dI.
dP/dI = (110(I^2 + I + 4) - 110I(2I + 1)) / (I^2 + I + 4)^2
= (110I^2 + 110I + 440 - 220I^2 - 110I) / (I^2 + I + 4)^2
= (110I^2 - 110I^2 + 110I - 110I + 440) / (I^2 + I + 4)^2
= 440 / (I^2 + I + 4)^2
2. Set the derivative equal to zero and solve for I:
440 / (I^2 + I + 4)^2 = 0
Since the numerator is nonzero, the derivative can never be zero. Therefore, there are no critical points where the derivative is zero.
3. Next, let's check for any vertical asymptotes or points where the derivative is undefined. These could potentially be maximum points.
The derivative will be undefined when the denominator of the fraction is equal to zero:
I^2 + I + 4 = 0
However, this equation has no real solutions, so there are no vertical asymptotes or points where the derivative is undefined.
Therefore, there are no critical points, and we cannot directly find the light intensity at which the photosynthesis rate P is a maximum using calculus.
To get a maximum value for P, you could try different values of I and evaluate the function P. Keep in mind that P represents the rate of photosynthesis, so you can compare the values of P to find the maximum.