radioactive element decays help please

Question

A radioactive element decays into non-radioactive substances. In 1000 days the radioactivity of a sample decreases by 5353 percent. Give your answer to the following questions round to the nearest whole number of days.

(a) What is the half-life of the element?

half-life:    (days)

(b) How long will it take for a sample of 100 mg to decay to 8484 mg?

time needed:    (days)

Answer 1

in 10 days, it decreases by 53 percent

47 % is left after 10 days

half life is when 50% is left, slightly less than 10 days

.5 = e^rt

.47 =e^10r

ln.47 = 10r

r = ln.47/10= about -.0755 = rate of decay per day

.5 = e^(ln.47/10)t

1/2 = e^.1tln.47

ln(1/2) = .1tln.47

.1t = ln.5/ln.47

t = 10ln.5/ln.47

t =about 9.1804827 days

or

.5 = e^-.0755t

ln.5 = -.0755t

t = ln.5/-.0755 = about 9.18 days

that looks about right, slightly less than 10 days

= about 9 days for half life

how long for 100 mg to decay to 84 mg?

.84 = e^-.0755t, solve for t

ln.84 = -.0755t

t = ln.84/-.0755

= about .23092 days

= about 6 hours

= about 0 days rounded to nearest number of days