Differential equations assignment
I conciliate be checking for structure, conceptual discernment, and suitable unrythmical despatch, as courteous as completion of the collections.
• Show as plenteous is-sue as you can, inhale sketches if expedient and evidently elucidate why you are doing what you are doing.
• Use rectify unrythmical notation.
• Show your is-sue vertically.
• You may is-sue behind a while your classmates. However, content surrender your own is-sue!
• Is-sue on a disconnected prevarication of article, and dedicate each collection evidently.
1. Carbon Dating An weighty machine in archeological inquiry is radiocarbon dating, familiar by the American chemist Willard F. Libby. This is a media of determining the age of convinced forest and fix sweepings, hence of constitutional esthetics such as carnal or civilized bones or artifacts fix buried at the corresponding rolls. Radiocarbon dating is established on the circumstance that some forest or fix sweepings comprehend residual measures of carbon-14 (C-14)−A radioactive isotope of carbon. This isotope is accumulated during the period of the fix and begins to decrease at its cessation. Since the half-life (i.e., the measecure of space it accepts for a measecure of radioactive esthetic to decrease to one-half of its primordial measure) of C-14 is very hanker (almost 5730 years!!!), measurable measures of C-14 sojourn behind multifarious thousands of years. If roll a little element of the primordial measecure of C-14 is stationary introduce, then by alienate laboratory measurements the symmetry of the primordial measecure of C-14 that sweepings can be accurately robust. In other opinion, if Q(t) is the measecure of C-14 at space t and Q 0 is the primordial measure, then the proportion Q(t)/Q 0 can be robust, at last if this measecure is not too little. (FYI: The introduce techniques encourage the use of this course for space periods of 50,000 years or more!)
(a) (3 tops) Assuming that Q satisﬁes the diﬀerential equation
dQ = −rQ , dt
ﬁnd an indication for Q(t) at any space t (in years), if Q(0) = Q 0 . That is, explain the corresponding IVP.
(b) (2 top) Determine the decrease invariable r for C-14 to THREE signiﬁcant digits. Also, what is the ace of this decrease invariable?
(c) (5 tops) Suppose that convinced sweepings are discovered in which the present residual measecure of C-14 is 25% of the primordial measure. Determine the age of these sweepings to the undeviating year.
2. Defiled Swimming Pool A swimming pool at a instil precinct comprehends 50,000 gallons of instil. Unfortunately, it has been defiled by 3,000 g of a nontoxic dye that leaves a swimmer’s bark an repultiive unfinished. The pool’s ﬁltering rule can accept instil from the pool, eject the dye, and reappear the instil to the pool at a ﬂow admonish of 250 gal/min.
(a) (4 tops) Write down the IVP for the ﬁltering process; let D(t) be the measecure of dye (in grams) in the pool at space t (in minutes). Behind fitness down the IVP, explain it!
(b) (2 tops) The instil precinct is scheduled to public in 3 hours. It has been robust that the eﬀect of the dye is insensible if its strain is near than 0.02 g/gal. Is the ﬁltering rule preferable of reducing the dye strain to this roll behind a whilein 3 hours?
(c) (2 tops) How hanker would it accept for the strain of dye to ﬁrst strain the estimate 0.02 g/gal?
(d) (2 tops) Find the ﬂow admonish (to the undeviating integer) that is suﬃcient to terminate the strain 0.02 g/gal behind a whilein 3 hours. Make secure to embrace rectify aces.