Question 2: (1 point) After drinking, the body eliminates 37% of the alcohol pre

Question

Question 2: (1 point) After drinking, the body eliminates 37% of the alcohol present in the body per hour. a) The amount of alcohol in grams in the body on an hourly basis is described by a discrete time dynamical system (DTDS) of the form n+1fn), where an is the number of grams of alcohol in the body after n hours. Give the updating function f (as a function of the variable ) Answer:fx)- b) Peter had three alcoholic drinks that brought the alcohol content in his body to 48 grams, and then he stopped drinking. Give the initial condition (in grams) for the DTDS in (a). Answer kograms : x0 = c) Find the solution of the DTDS in (a) with the initial condition given in (b). (Your answer will be a function of the variable n, which represents time in hours.) Answer: xn = - d) If the amount of alcohol in Peter's body has to be below 8 grams before one can drive, how long (in hours) does Peter have to wait before he can drive? Round up to the nearest integer value. Answer: _ hours

Explanation / Answer

a). If xn is the no. of grams of alcohol in the body after n hours , then xn+1= 0.73xn.

b).x0 = 48 grams.

c). The DTDS function is xn+1= 0.73xn . It is a geometric series with 48 as the 1st term and 0.73 as the common ratio. The nth term is xn =48*(0.73)n-1.

d). If xn =48*(0.73)n-1 < 8, then (0.73)n-1 < 8/48 = 1/6. On taking log of both the sides, we have (n-1)log 0.73 < log 1/6 or, -(n-1)*0.136677139 < -0.77815125 or, n-1 > 0.77815125/0.136677139 or, n-1> 5.69 so that n > 6.69 hours ( on rounding off to 2 decimalo places). Thus, Peter has to wait for 6.69 hours.

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