In many population growth problems, there is an upper limit beyond which the population cannot grow. many scientists agree that the earth will not support a population of more than 16 billion. there were 2 billion people on earth in 1925 and 4 billion in 1975. if is the population years after 1925, an appropriate model is the differential equation
dy/dt=ky(16-y)

note that the growth rate approaches zero as the population approaches its maximum size. when the population is zero then we have the ordinary exponential growth described by y'=16ky. as the population grows it transits from exponential growth to stability.

(a) solve this differential equation.
(b) the population in 2015 will be
(c) the population will be 9 billion some time in the year

Nthe opening scene of the movie raiders of the lost ark, indiana jones replaces a golden statue with a bag of sand. the platform on which the statue is placed is designed to detect the mass of the statue. if the bag of sand has a different mass than the statue, then a mechanism triggers and destruction ensues. the density of gold is about 20\text{ g/cm}^320 g/cm 3 20, space, g, slash, c, m, start superscript, 3, end superscript. the density of sand can be estimated at 2.5\text{ g/cm}^32.5 g/cm 3 2, point, 5, space, g, slash, c, m, start superscript, 3, end superscript. the gold statue can be modeled as a cylinder with height 25\text{ cm}25 cm25, space, c, m and radius 8\text{ cm}8 cm8, space, c, m. in terms of \piπpi, what is the mass of sand indiana jones needs to avoid triggering the mechanism?