Question 277974
half life = 578 hours.


exponential decay is assumed.


this means that 1/2 = 1 * (1+g)^t where g = the growth rate and t = time in hours.


.5 = (1+g)^t


since t = 578, then this equation becomes:


.5 = (1+g)^578


take the log of both sides of this equation to get:


log(.5) = log((1+g)^578)


this is equivalent to:


log(.5) = 578 * log(1+g)


divide both sides of this equation by 578 to get:


log(1+g) = log(.5) / 578 which becomes:


log(1+g) = -.000520814


this means that 1+g = .998801502 which means that g = .998801502 - 1 = -.001198498


to see if this is correct, substitute in original equation to see if it is true.


original equation is:


.5 = (1+g)^578


this becomes:


.5 = .998801502^578 = .5 confirming the value of g is good.


now that we know the value of (g), we should be able to answer the other questions.


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a) If a sample has an initial mass of 64 mg, a function that models the mass in mg after t hours is:


a(t) = 64 * (.998801502)^t


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b) The amount remaining after 75 hours will be about:


a(t) = 64 * (.998801502)^75 = 58.49503274


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c) The initial mass will decay to 12 mg after about:


a(t) = 64 * (.998801502)^t becomes:


12 = 64 * (.998801502)^t


divide both sides by 64 to get:


12/64 = (.998801502)^t


take log of both sides of the equation to get:


log(12/64) = log(.998801502^t) which becomes:


log(12/64) = t*log(.998801502) 


divide both sides of this equation by log(.998801502) to get:


t = log(12/64)/log(.998801502) to get:


t = 1395.891675 hours.


the sample will decay to 12 mg in approximately 1395.89 hours.


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extra:


if the function is correct, then the 64 mg should decay to 32 mg in 578 hours.


formula is:


32 = 64 * .998801502^t


divide both sides of this equation by 64 to get:


32/64 = .998801502^t


take log of both sides of this equation to get:


log(32/64) = log(.998801502^t) which becomes:


log(32/64) = t*log(.998801502).


divide both sides of this equation by log(.998801502) to get:


t = log(32/64)/log(.998801502) which becomes:


t = 578


any minor discrepancy in the number is due to rounding.