Question
Find the Orthogonal Trajectories of the family of curves: F = {e^x - e^-y = A: A elementof R} Find the Orthogonal Trajectories of the family of curves: F_1 = {y(x^2 + C) - 2 = 0: C elementof R} Consider a colony of E. Coli bacteria that is growing exponentially. A microbiologist finds that, initially, 1,000 bacteria are present and 50 minutes later there are 10,000 bacteria. a) Find an expression for the number of bacteria Q(t) after t minutes. b) When will there be 1,000,000 bacteria? A 1,000 gallon tank containing 500 gallons of a solution in which 50 pounds of salt is dissolved. A solution containing 4 lb/gal so salt is pumped into the tank at a rate of 4 gal/min. The mixture is allowed to flow out of the tank at a rate of 2 gal/min. Assuming that the mixture is perfectly stirred at all times, how much salt will be in the tank when it is full? The coroner arrives at the scene of a murder at 3 a.m. He takes the temperature of the body and find it to be 50.6 degree. He waits 1 hour, takes the temperature again, and finds it to be 48.2 degree. The body is in a meat freezer, where the temperature is a constant 10 degree. When was the murder committed? At small military base housing 1,000 troops, each of whom is susceptible to a certain virus inflection. Assuming that the during the course of the epidemic the rate of change (with respect to time) of the number of infected troopers is jointly proportional to then number of troopers inflected and the number of uninfected troopers. If at the initial outbreak, there was one trooper infected and 2 days later there were 5 troopers infected, express the number of infected troopers as a function of time.
Explanation / Answer
1) e^x - e^(-y) = A
e^x + e^(-y) y' = 0
y' = -e^x /e^(-y)
y' = f(x,y)
for orthogonal we solve
y' = -1/ f(x,y)
y' = e^(-y)/e^x
e^y dy = e^(-x) dx
e^y = -e^(-x) + c
hence
orthogonal trajectoty is
e^y = c - e^(-x)
solve Q2 similarly
3) P(t) = P0 e^(rt)
P0 = 1000 , at t= 50 , P(50) = 10000
hence 10 = e^(50 *r)
r = ln (10)/50 =0.046051701
P(t) = 1000 e^(0.046051701* t) where t is in minutes
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