Jessica and matthew are running toward the point p along the straight paths that make a fixed angle of θ. suppose that matthew runs with velocity va meters per second and jessica with velocity vb meters per second. assume that θ = π/3. use the chain rule to determine the rate at which the distance f between matthew and jessica is changing when x = 40, y = 30, va = 4 m/s, and vb = 3 m/s.

54 degrees incident angle = reflected angle

angle is from normal turned to line of ray

bacteria that live on the ocean floor are sustained by phytoplankton. phytoplankton, microscopic marine photosynthetic organisms, have a vastly significant role to play not only in the marine food web of which they’re part of, but also on a more global scale. despite their infinitely small size in comparison to other marine organisms, these tiny creatures occupy an immensely important ecological niche: they are the foundation of the marine food web, and as primary producers, play key roles in supporting all other organisms in the marine environment, as well as in the regulation of the earth’s climate through the sequestration of carbon, oxygen production, and other related processes. phytoplankton account for roughly half of all global primary productivity; therefore, their significance extends far beyond the marine environment alone. there is an intriguing sense of irony in the realization that these tiny living beings that often live out their existence unnoticed and undetected by the rest of the world, have such a far-reaching impact on the lives of virtually all other living organisms on the planet, particularly on those in the marine environment. the world that we have become accustomed to has been and is continuously shaped by the workings of these miniscule yet vital “plants of the sea”.

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