A culture of cells in a lab has a population of 100 cells when nutrients are added at time t = 0. Suppose the population N(t) (in cells/hr) increases at arate given by  N'(t) = 90e-0.1t.Find N(t), for t ≥ 0.

18. A rectangle with length l cm, and height h cm, is inscribed in a circle of radius 10 cm, as shown in the adjacent diagram. The radius of the circle does not change, but the length the rectangle is decreasing at the 10 constant rate of 8 cm·s-1. Let A be the area of the rectangle (measured in cm2). (a) Show that A = lV400 – l².

One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. Let P represent the number of female insects in a population and S the number of sterile males introduced each generation. Let r be the per capita rate of production of females by females, provided that their chosen mate is not sterile. Then the female population is related to time t by t = ∫(P + S)dP/P[(r-1)P - S]   Suppose an insect population with 10,000 females grows at a rate of r = 1.2 and 900 sterile males are added initially. Evaluate the integral to give an equation relating the female population to time. (Note that the resulting equation cannot be solved explicitly for P.)

A sample contains 560 grams of Uranium-232, a radioactive isotope, that decays according to the function N(t) = 560e-0.010060t,   where t is in years.Find the rate at which the Uranium-232 is decaying when t = 20 years. (Find N'(20))Round your answer to two decimal places.