Solution Found!
Photosynthesis Much of the earths photosynthesis occurs in the oceans. The rate of
Chapter 5, Problem 62(choose chapter or problem)
Photosynthesis Much of the earth’s photosynthesis occurs in the oceans. The rate of primary production (as discussed in Exercise 61) depends on light intensity, measured as the flux of photons (that is, number of photons per unit area per unit time). For monochromatic light, intensity decreases with water depth according to Beer’s Law, which states that \(I(x)=e^{-k x}\), where x is water depth. A simple model for the relationship between rate of photosynthesis and light intensity is \(P(I)=a I\), where a is a constant and P is measured as a mass of carbon fixed per volume of water, per unit time.
(a) What is the rate of photosynthesis as a function of water depth?
(b) What is the total rate of photosynthesis of a water column that is one unit in surface area and four units deep?
(c) What is the total rate of photosynthesis of a water column that is one unit in surface area and x units deep?
(d) What is the rate of change of the total photosynthesis with respect to the depth x?
Questions & Answers
QUESTION:
Photosynthesis Much of the earth’s photosynthesis occurs in the oceans. The rate of primary production (as discussed in Exercise 61) depends on light intensity, measured as the flux of photons (that is, number of photons per unit area per unit time). For monochromatic light, intensity decreases with water depth according to Beer’s Law, which states that \(I(x)=e^{-k x}\), where x is water depth. A simple model for the relationship between rate of photosynthesis and light intensity is \(P(I)=a I\), where a is a constant and P is measured as a mass of carbon fixed per volume of water, per unit time.
(a) What is the rate of photosynthesis as a function of water depth?
(b) What is the total rate of photosynthesis of a water column that is one unit in surface area and four units deep?
(c) What is the total rate of photosynthesis of a water column that is one unit in surface area and x units deep?
(d) What is the rate of change of the total photosynthesis with respect to the depth x?
ANSWER:Step 1 of 4
a)
Substitute for \(I\) with \(e^{-k x}\) to turn it into a function of water depth, x.
\(\begin{aligned}
P(I) & =a I \\
I(x) & =e^{-k x} \\
P(x) & =a e^{-k x}
\end{aligned}\)