/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 In light-dependent photosynthesi... [FREE SOLUTION] | 91Ó°ÊÓ

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In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of a sample of photosynthetically active radiations (PAR) is measured to the nearest nanometer. The red range is \(675-700 \mathrm{nm}\) and the blue range is \(450-500 \mathrm{nm}\). Let \(A\) denote the event that PAR occurs in the red range and let \(B\) denote the event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events: (a) \(A\) (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

Short Answer

Expert verified
Sample space: 400 nm to 700 nm; (a) 675-700 nm; (b) 450-500 nm; (c) Empty set; (d) 450-500 nm and 675-700 nm.

Step by step solution

01

Understanding the Sample Space

The sample space for photosynthetically active radiation (PAR) consists of all possible wavelengths that PAR can take, measured to the nearest nanometer. This range is from 400 nm to 700 nm, covering all wavelengths that plants use for photosynthesis.
02

Defining Event A

Event \( A \) occurs when the wavelength is in the red range, specifically from 675 nm to 700 nm. These are wavelengths that are absorbed during photosynthesis.
03

Defining Event B

Event \( B \) occurs when the wavelength is in the blue range, from 450 nm to 500 nm. Wavelengths in this range are also absorbed by plants and play a crucial role in photosynthesis.
04

Understanding Event A ∩ B

The event \( A \cap B \) represents the intersection of events \( A \) and \( B \). This would mean finding wavelengths that are both in the red and blue ranges. Since there is no overlap between 675 nm to 700 nm and 450 nm to 500 nm, \( A \cap B \) is the empty set \( \emptyset \).
05

Understanding Event A ∪ B

The event \( A \cup B \) represents the union of events \( A \) and \( B \). This includes all wavelengths that are either in the red range (675 nm to 700 nm) or the blue range (450 nm to 500 nm), effectively combining the two ranges with no overlap.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Photosynthetically Active Radiation
Photosynthetically Active Radiation (PAR) includes the range of light wavelengths that plants can use for photosynthesis.
This range spans from 400 nanometers (nm) to 700 nm. Within this spectrum, plants efficiently convert light energy into chemical energy. The PAR range is vital because it contains the wavelengths that chlorophyll, the green pigment in plants, absorbs to fuel photosynthesis.
Chlorophyll primarily absorbs light in the blue and red wavelengths and reflects green, which is why plants appear green. Understanding PAR is critical when studying how plants harness light.
Knowing which wavelengths fall under PAR can help in optimizing growth conditions in agriculture or research. For example, when growing plants indoors using artificial lighting, targeting PAR specifically can enhance plant growth and reduce energy waste.
Light Wavelengths
Light wavelengths refer to the different segments of the electromagnetic spectrum that are visible to the human eye and absorbed by plants for photosynthesis. Key Wavelength Ranges Essential for Photosynthesis:
  • Blue Range (450-500 nm)
  • Red Range (675-700 nm)
The blue and red ranges are crucial for photosynthesis because they drive essential processes within the plant, such as the production of ATP and NADPH, which are energy carriers. Each wavelength range plays a specific role:
  • Blue light supports vegetative growth, influencing leaf development and increasing the rate of photosynthesis.
  • Red light helps in flowering and fruiting, pivotal for reproductive phases.
Understanding how different wavelengths affect plant processes can inform decisions in agriculture, such as crop lighting strategies or plant breeding.
Event Intersection and Union
When discussing probabilities in photosynthesis, specifically with PAR, two mathematical concepts come into play: intersection and union of events. Event Intersection (\(A \cap B\)) occurs when there is a common element between two sets. In the context of PAR, this means finding a wavelength that is both in the red and blue wavelength ranges.
However, given the non-overlapping nature of the red (675-700 nm) and blue (450-500 nm) ranges, the intersection \(A \cap B\) results in an empty set \(\emptyset\).Event Union (\(A \cup B\)) includes all elements that are in either set or both. In photosynthesis, this means any wavelength in the red or blue range.
For PAR, this union is the set of all wavelengths from 450 to 500 nm and from 675 to 700 nm. Understanding these concepts aids in visualizing how light characteristics can overlap or be distinct in affecting plant processes. It also broadens insights into spectral efficiencies in various environmental or artificial conditions.

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