a) There are 5 organelles found in a root hair cell. The 5 organelles are the nucleus, plasma membrane, cytoplasm, vacuole and the cell wall.
b) The functions of the cell is to absorb water and mineral ions from soil. The projecting section has large surface area for absorption of water and minerals from soil water, there are no chloroplasts. The nucleus stores the DNA of the cell and it controls all the activities in the cell. Cellular activities takes place in the cytoplasm. The cell membrane controls the movement of substance in and out of the cell. It is a selectively permeable membrane because it allows only some substance to pass through. The large central vacuole is filled with a fluid (sap fluid: contain dissolved sugars and amino acid) and is surrounded by a sap membrane or tonoplast. It provides structural support and maintains turgor pressure. The cell wall gives the plant its shape and has a fully permeable structure (allows all substance to pass through).
c) Root hairs form an important surface over which plants absorb m
ost of their water and nutrients. They are also directly involved in the formation of root nodules in legume plants.
They have a large surface area, relative to the other cells, that helps them absorb water and minerals more efficiently.
Information sourced from: Wikipedia, reference papers
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Pictures of root hair cells: