The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function
P =
100I / I2 + I + 9
where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?
Can't make out your equation, is it
P = 100I / (I^2 + I + 9) ?
To find the light intensity at which P is a maximum, we need to find the maximum point of the function. This corresponds to finding the critical points of the function and determining whether they are maximum or minimum points.
Step 1: Take the derivative of the function P with respect to I.
The derivative of P with respect to I can be found using the quotient rule of differentiation:
P' = (I^2 + I + 9)(100) - (100I)(2I + 1) / (I^2 + I + 9)^2
= (100I^2 + 100I + 900 - 200I^2 - 100I) / (I^2 + I + 9)^2
= (-100I^2 + 0I + 900) / (I^2 + I + 9)^2
= 900 / (I^2 + I + 9)^2 - 100 / (I^2 + I + 9)
Step 2: Set the derivative equal to zero and solve for I.
Setting P' = 0, we have:
900 / (I^2 + I + 9)^2 - 100 / (I^2 + I + 9) = 0
To simplify the equation, we can multiply both sides by (I^2 + I + 9)^2 to eliminate the denominators:
900 - 100(I^2 + I + 9) = 0
900 - 100I^2 - 100I - 900 = 0
Simplifying further:
-100I^2 - 100I = 0
-100I(I + 1) = 0
From this, we have two possibilities:
1. -100I = 0 (giving I = 0)
2. I + 1 = 0 (giving I = -1)
Step 3: Determine whether the critical points are maximum or minimum points.
To determine the nature of the critical points, we can use the second derivative test. The second derivative of P with respect to I can be found by taking the derivative of P', which is P":
P" = dP'/dI = (-200I(I + 1)(I^2 + I + 9)^2 - 900(2I + 1)(I^2 + I + 9)) / (I^2 + I + 9)^4
Evaluate P" at the critical points we found:
At I = 0, P" = (-900(2(0) + 1)(0^2 + 0 + 9)) / (0^2 + 0 + 9)^4 = -900 / 9^4 < 0
At I = -1, P" = (-200(-1)(-1 + 1)((-1)^2 + (-1) + 9)^2 - 900(2(-1) + 1)((-1)^2 + (-1) + 9)) / ((-1)^2 + (-1) + 9)^4 = 0 / 9^4 = 0
Since P" < 0 at I = 0, we have a maximum point at I = 0. Therefore, for a light intensity of 0 thousands of foot-candles, P is a maximum.