Suppose a small cannonball weighing 20 pounds is shot vertically upward, with an initial velocity v0 = 260 ft/s. the answer to the question "how high does the cannonball go? " depends on whether we take air resistance into account. if air resistance is ignored and the positive direction is upward, then a model for the state of the cannonball is given by d2s/dt2 = −g (equation (12) of section 1.3). since ds/dt = v(t) the last differential equation is the same as dv/dt = −g, where we take g = 32 ft/s2. if air resistance is incorporated into the model, it stands to reason that the maximum height attained by the cannonball must be less than if air resistance is ignored. (a) assume air resistance is proportional to instantaneous velocity. if the positive direction is upward, a model for the state of the cannonball is given by m dv dt = −mg − kv, where m is the mass of the cannonball and k > 0 is a constant of proportionality. suppose k = 0.0025 and find the velocity v(t) of the cannonball at time t. use the result obtained in part (a) to determine the height s(f) of the cannonball measured from ground level. find the maximum height attained by the cannonball. (round your answer to two decimal places.)

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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