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5. Let's take one more step in improving this population model. Assume that there is a maximum numberM thata population can sustain in orderto survive over manyr generations. After all. there are limited resources in our world. When the population size is much smaller than M, resources playr a negli'ble role in determining the population growth rate. When the population size exoeeds M. it experiences a negative growth rate. When the population size equals M, the population growth rate equals zero. a] Show using notation what the paragraph above explains in three separate cases {when P s: M. when P 3:- M, andwhenP = M]. b] Design a factor that will achieve this effect and write out the improved differential equation. c] Draw the family of curves that represent P in relation to t. o. Radioactive substances decav at a rate proportional to their remaining mass M. a] Express this rate of decayr as a differential equation. b] 1vl'lu'hen a living organism dies it oeases to replaoe the carbon isotope (2-14, and the [114 that is present decays with a halflife of about STED years. lf archeologists discover a fossilized bone that has 30% of the C- 14 of a live bone, approximatelyr how old is it